(a OR b) -> c
= ~(a OR b) OR c
= (~a AND ~b) OR c
= (~a OR c) AND (~b OR c)
= (a -> c) AND (b -> c) as required
I haven’t formally learnt logic so I’m not sure if my proof is what you’d call rigorous, but the result is pretty useful for splitting up conditionals in proofs like some of the number theory proofs I’ve been trying. E.g.
Show that if a is greater than 2 and a^m + 1 is prime, then a is even and m is a power of 2
In symbolic form this is:
∀a >= 2 ( a^m + 1 is prime -> a is even AND m is a power of 2 )
The contrapositive is:
∀a >= 2 ( a is odd OR m is NOT a power of 2 -> a^m + 1 is composite )
and due to the result above, this becomes
∀a >= 2 ( a is odd -> a^m + 1 is composite ) AND ( m is NOT a power of 2 -> a^m + 1 is composite )
so you can just prove two simpler conditionals instead of one more complicated one.
So observing that this looks true because a OR b only fails when both a and b do, here’s an alternative:
a OR b -> c
Conditional contraposition:
~c -> ~(a OR b)
De Morgan’s law 1:
~c -> ~a AND ~b
I’m not actually sure what this kind of distribution is called, but it makes sense:
(~c -> ~a) AND (~c -> ~b)
Two more contrapositions:
(a -> c) AND (b -> c)