Theorem wl-orel12

Description: In a conjunctive normal form a pair of nodes like ( ph \/ ps ) /\ ( -. ph \/ ch ) eliminates the need of a node ( ps \/ ch ) . This theorem allows simplifications in that respect. (Contributed by Wolf Lammen, 20-Jun-2020)

	Ref	Expression
Assertion	wl-orel12	|- ( ( ( ph \/ ps ) /\ ( -. ph \/ ch ) ) -> ( ps \/ ch ) )

Proof

Step	Hyp	Ref	Expression
1		pm2.1	|- ( -. ph \/ ph )
2		orel1	|- ( -. ph -> ( ( ph \/ ps ) -> ps ) )
3		orc	|- ( ps -> ( ps \/ ch ) )
4	2 3	syl6com	|- ( ( ph \/ ps ) -> ( -. ph -> ( ps \/ ch ) ) )
5		notnot	|- ( ph -> -. -. ph )
6		orel1	|- ( -. -. ph -> ( ( -. ph \/ ch ) -> ch ) )
7	5 6	syl	|- ( ph -> ( ( -. ph \/ ch ) -> ch ) )
8		olc	|- ( ch -> ( ps \/ ch ) )
9	7 8	syl6com	|- ( ( -. ph \/ ch ) -> ( ph -> ( ps \/ ch ) ) )
10	4 9	jaao	|- ( ( ( ph \/ ps ) /\ ( -. ph \/ ch ) ) -> ( ( -. ph \/ ph ) -> ( ps \/ ch ) ) )
11	1 10	mpi	|- ( ( ( ph \/ ps ) /\ ( -. ph \/ ch ) ) -> ( ps \/ ch ) )