Theorem consensus 574

Description: The consensus theorem. This theorem and its dual (with \/ and /\ interchanged) are commonly used in computer logic design to eliminate redundant terms from Boolean expressions. Specifically, we show the term (ps /\ ch) on the left-hand side is redundant.

Assertion

Ref	Expression
consensus	|- ((((ph /\ ps) \/ (-. ph /\ ch)) \/ (ps /\ ch)) <-> ((ph /\ ps) \/ (-. ph /\ ch)))

Proof of Theorem consensus

Step	Hyp	Ref	Expression
1		id 9	. . 3 |- (((ph /\ ps) \/ (-. ph /\ ch)) -> ((ph /\ ps) \/ (-. ph /\ ch)))
2		dedlema 569	. . . . . . 7 |- (ph -> (ps <-> ((ps /\ ph) \/ (ch /\ -. ph))))
3	2	biimpd 135	. . . . . 6 |- (ph -> (ps -> ((ps /\ ph) \/ (ch /\ -. ph))))
4	3	adantrd 308	. . . . 5 |- (ph -> ((ps /\ ch) -> ((ps /\ ph) \/ (ch /\ -. ph))))
5		dedlemb 570	. . . . . . 7 |- (-. ph -> (ch <-> ((ps /\ ph) \/ (ch /\ -. ph))))
6	5	biimpd 135	. . . . . 6 |- (-. ph -> (ch -> ((ps /\ ph) \/ (ch /\ -. ph))))
7	6	adantld 307	. . . . 5 |- (-. ph -> ((ps /\ ch) -> ((ps /\ ph) \/ (ch /\ -. ph))))
8	4, 7	pm2.61i 110	. . . 4 |- ((ps /\ ch) -> ((ps /\ ph) \/ (ch /\ -. ph)))
9		ancom 333	. . . . 5 |- ((ps /\ ph) <-> (ph /\ ps))
10		ancom 333	. . . . 5 |- ((ch /\ -. ph) <-> (-. ph /\ ch))
11	9, 10	orbi12i 216	. . . 4 |- (((ps /\ ph) \/ (ch /\ -. ph)) <-> ((ph /\ ps) \/ (-. ph /\ ch)))
12	8, 11	sylib 173	. . 3 |- ((ps /\ ch) -> ((ph /\ ps) \/ (-. ph /\ ch)))
13	1, 12	jaoi 275	. 2 |- ((((ph /\ ps) \/ (-. ph /\ ch)) \/ (ps /\ ch)) -> ((ph /\ ps) \/ (-. ph /\ ch)))
14		orc 225	. 2 |- (((ph /\ ps) \/ (-. ph /\ ch)) -> (((ph /\ ps) \/ (-. ph /\ ch)) \/ (ps /\ ch)))
15	13, 14	impbi 139	1 |- ((((ph /\ ps) \/ (-. ph /\ ch)) \/ (ps /\ ch)) <-> ((ph /\ ps) \/ (-. ph /\ ch)))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   \/ wo 195   /\ wa 196

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198

metamath.org