Is it possible to prove conventional current is always equivalent to actual current? I understand how the conventional current is logically equivalent to the actual current of electrons in a circuit. However, whenever I'm studying some new concept, and things are assumed as working with a current of protons (or positively charged particles), I have to always prove to myself that this is true because it also holds true for a current of electrons.
This proving things to myself is getting irritating.
I wanted to know if there is an argument, a very fundamental one, that could make me stop wanting to prove these things to myself? Something like, “whatever we can say about a current of positive charge particles flowing one way, we can say about electrons flowing the opposite way because ....[followed by a proof]”.
 A: I suggest you should look at all your arguments to date and see what they have in common. I'd almost be willing to bet that they can be reduced to something like the following:
The equations governing a circuit's behaviour are linear in the vector of state variables, amongst whose members is the current. That is, if $\vec{U}$ is a column vector of state variables and it solves the equations, then so does $\alpha \vec{U}$ as well, where $\alpha$ is any real (or complex, in the case of phasor notation) scalar. This linearity follows from the linearity of all the circuit elements you use together with the linearity of the equations used to combine them: check that the relationships between the state variables defining an element's electrical behaviour holds in each case. For example, the inductor is defined by $v(t) = - L \,\mathrm{d}_t i(t)$, and this equation holds true if we make the transformation $(v,\,i)\mapsto(\alpha\,v,\,\alpha\,i)$. Then you check the linearity of the equations that combine these equations in a circuit. These are simply the Kirchoff voltage and current laws (conservation of energy around a loop and charge, respectively), and are most decidedly linear in the $v$ and $i$ variables they combine.
The arbitrary choice of current direction is simply this argument in the special case where $\alpha=-1$. You must switch the sign of all the voltage definitions together withyour currents.
Actually equations of state that survive multiplication by $-1$ are more general than simply linear ones, but this is the easiest argument. Note that the above arguments do not hold for all physics governing the electrical circuit: protons going the opposite way to the equivalent number of electrons going the other are most certainly not the same physics. A good example was given to you by user dmckee:

Note that the Hall Effect unambiguously differentiates between the two cases, so it is not true that “whatever we can say about a current of positive charge particles flowing one way, we can say about electrons flowing to opposite way", but for the usual questions of circuit analysis it doesn't make any difference. 

A: From Magnet University

You may follow an imagined direction of current (conventional flow) or the actual (electron flow) with equal success insofar as circuit analysis is concerned. Concepts of voltage, current, resistance, continuity, and even mathematical treatments such as Ohm's Law and Kirchhoff's Laws remain just as valid with either style of notation.

A: As Rod notes in his answer, any purely linear circuit will certainly behave the same way if all the currents (and voltages) are scaled by any (non-zero) constant value $\alpha$, including the case $\alpha = -1$.
In fact, a typical feature of linear circuit elements, like resistors, inductors and (non-electrolytic) capacitors, is that they'll work equally well regardless of which way you plug them in.  If all your circuit elements are like that, it should not be hard to show that your entire circuit will also have the same property.
That said, there are also plenty of common circuit elements that are not linear, and which cannot simply be reversed without changing their behavior — the obvious example being diodes.  An ideal diode will freely conduct current in one direction, but block any current flowing the other way, making it about as asymmetric as a component can be.  Thus, a circuit containing a diode will not behave the same way under current reversal.
Fortunately, when we write out specs for non-linear components like diodes, we (always!) take into account the fact that conventional current goes the opposite way from electron flow.  Thus, when a diode is marked as conducting current in one direction, what that actually means is that it lets electrons pass in the opposite direction.
So, effectively, the reason why conventional currents work for non-linear circuit elements is that we've defined a conventional current of $I$ amperes in one direction to mean a flux of 6.241 × 1018 electrons per second in the opposite direction.  All the specifications for such circuit elements take this definition into account, so that when you look at the current–voltage graph for such an element to determine its voltage response, the axes are consistently labeled in amperes and volts of conventional current and voltage, respectively.
As for why linearity also matters, it's because wires are linear components, and so Kirchhoff's laws are linear.  Thus, when we wish to calculate how currents flow between components in a circuit, it doesn't matter how we choose to scale the currents and voltages (or, in particular, which way we choose to have the current flow).
