(p -> r) is equivalent to (殆 v r) i.e p implies r means that r is true or p is not true.

(p v q) ^ (殆 v r) ^ (段 v r) and (p ^ 殆) v (r ^ q) ^ (段 v r) both mean that p or q is true, and r is true or p is false, and r is true or q is false.

(p ^ 殆) is false hence (r ^ q) ^ (段 v r) is true, and r and q being true implies r is true.

From: Ashraf Samhouri <discussions@mathforum.org>
To: discretemath@mathforum.org
Sent: Monday, 19 March 2012, 8:51
Subject: Tautology Proof Question

Hello guys,

While studying, I've passed through this proof (in the class notes):

Example:
Proof that:
( (p v q) ^ (p -> r) ^ (q -> r) ) -> r
is a tautology?

The solution is:
( (p v q) ^ (殆 v r) ^ (段 v r) ) -> r
( (p ^ 殆) v (r ^ q) ^ (段 v r) ) -> r
(F v (q ^ 段) v r) -> r
r -> r
毒 v r
.: T

Actually I can't understand the steps, am not sure if the notes are
well-written here, but I'll appreciate any help in explaining the
proof.

Regards,
ASH