Equations in physics are often written without quantifiers. For instance, from time to time we can see the equation $$E = mc^2$$ is casually written down.
I am wondering shouldn't it be written rigorously as something like: For all $E$ in a given range of values and all $m$ in a given range of values, we have $E =mc^2$? If not, then logically the equation $E = mc^2$ is sometimes valid and sometimes invalid as $E, m$ run through their given ranges, so that one cannot use "$E = mc^2$" to express what it is intended to mean.
Then how does one incorporate this phys-mathematics writing style, as illustrated in the beginning, with the authentic mathematical writing style, as illustrated in the second paragraph, in the sense of logical rigor?