Theorem wl-orel12 31893

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.)

Assertion

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

Proof of Theorem wl-orel12

Step	Hyp	Ref	Expression
1		pm2.1 423	. 2 |- ( -. ph \/ ph )
2		orel1 388	. . . 4 |- ( -. ph -> ( ( ph \/ ps ) -> ps ) )
3		orc 391	. . . 4 |- ( ps -> ( ps \/ ch ) )
4	2, 3	syl6com 36	. . 3 |- ( ( ph \/ ps ) -> ( -. ph -> ( ps \/ ch ) ) )
5		notnot1 127	. . . . 5 |- ( ph -> -. -. ph )
6		orel1 388	. . . . 5 |- ( -. -. ph -> ( ( -. ph \/ ch ) -> ch ) )
7	5, 6	syl 17	. . . 4 |- ( ph -> ( ( -. ph \/ ch ) -> ch ) )
8		olc 390	. . . 4 |- ( ch -> ( ps \/ ch ) )
9	7, 8	syl6com 36	. . 3 |- ( ( -. ph \/ ch ) -> ( ph -> ( ps \/ ch ) ) )
10	4, 9	jaao 516	. 2 |- ( ( ( ph \/ ps ) /\ ( -. ph \/ ch ) ) -> ( ( -. ph \/ ph ) -> ( ps \/ ch ) ) )
11	1, 10	mpi 20	1 |- ( ( ( ph \/ ps ) /\ ( -. ph \/ ch ) ) -> ( ps \/ ch ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 374   /\ wa 375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377

This theorem is referenced by:  wl-cases2-dnf  31894