Theorem bj-consensus 32562

Description: Version of consensus 999 expressed using the conditional operator. (Remark: it may be better to express it as consensus 999, using only binary connectives, and hinting at the fact that it is a Boolean algebra identity, like the absorption identities.) (Contributed by BJ, 30-Sep-2019.)

Assertion

Ref	Expression
bj-consensus	|- ( (if- ( ph , ps , ch ) \/ ( ps /\ ch ) ) <-> if- ( ph , ps , ch ) )

Proof of Theorem bj-consensus

Step	Hyp	Ref	Expression
1		anifp 1020	. . 3 |- ( ( ps /\ ch ) -> if- ( ph , ps , ch ) )
2	1	bj-jaoi2 32557	. 2 |- ( (if- ( ph , ps , ch ) \/ ( ps /\ ch ) ) -> if- ( ph , ps , ch ) )
3		orc 400	. 2 |- (if- ( ph , ps , ch ) -> (if- ( ph , ps , ch ) \/ ( ps /\ ch ) ) )
4	2, 3	impbii 199	1 |- ( (if- ( ph , ps , ch ) \/ ( ps /\ ch ) ) <-> if- ( ph , ps , ch ) )

Syntax hints:   <-> wb 196   \/ wo 383   /\ wa 384  if-wif 1012

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 197  df-or 385  df-an 386  df-ifp 1013

This theorem is referenced by: (None)