When proving that $ab = 0 \implies a = 0 \,\mbox{ or }\,b = 0$ for members $a$ and $b$ of a field, I used an argument like
- Suppose $ab = 0$ and $a \ne 0$ ... then $b = 0$.
- Now suppose $ab = 0$ and $b \ne 0$ ... then $a = 0$.
- Therefore, if $ab = 0$, then $a = 0$ or $b = 0$.
The general form of that argument would, as far as I can tell, be
$ (p \land \lnot q \to r) \land (p \land \lnot r \to q) \to (p \to q \lor r) $
Is that general form indeed a valid argument? How can I know for sure? (Is there a "for sure"?)