Theorem consensus 925

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 prove that the term (ps /\ ch) on the left-hand side is redundant. (Contributed by NM, 16-May-2003.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 20-Jan-2013.)

Assertion

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

Proof of Theorem consensus

Step	Hyp	Ref	Expression
1		id 19	. . 3 |- (((ph /\ ps) \/ (-. ph /\ ch)) -> ((ph /\ ps) \/ (-. ph /\ ch)))
2		orc 374	. . . . 5 |- ((ph /\ ps) -> ((ph /\ ps) \/ (-. ph /\ ch)))
3	2	adantrr 697	. . . 4 |- ((ph /\ (ps /\ ch)) -> ((ph /\ ps) \/ (-. ph /\ ch)))
4		olc 373	. . . . 5 |- ((-. ph /\ ch) -> ((ph /\ ps) \/ (-. ph /\ ch)))
5	4	adantrl 696	. . . 4 |- ((-. ph /\ (ps /\ ch)) -> ((ph /\ ps) \/ (-. ph /\ ch)))
6	3, 5	pm2.61ian 765	. . 3 |- ((ps /\ ch) -> ((ph /\ ps) \/ (-. ph /\ ch)))
7	1, 6	jaoi 368	. 2 |- ((((ph /\ ps) \/ (-. ph /\ ch)) \/ (ps /\ ch)) -> ((ph /\ ps) \/ (-. ph /\ ch)))
8		orc 374	. 2 |- (((ph /\ ps) \/ (-. ph /\ ch)) -> (((ph /\ ps) \/ (-. ph /\ ch)) \/ (ps /\ ch)))
9	7, 8	impbii 180	1 |- ((((ph /\ ps) \/ (-. ph /\ ch)) \/ (ps /\ ch)) <-> ((ph /\ ps) \/ (-. ph /\ ch)))

Syntax hints:  -. wn 3   <-> wb 176   \/ wo 357   /\ wa 358

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 177  df-or 359  df-an 360

This theorem is referenced by: (None)