Theorem anifp 1020

Description: The conditional operator is implied by the conjunction of its possible outputs. Dual statement of ifpor 1021. (Contributed by BJ, 30-Sep-2019.)

Assertion

Ref	Expression
anifp	|- ( ( ps /\ ch ) -> if- ( ph , ps , ch ) )

Proof of Theorem anifp

Step	Hyp	Ref	Expression
1		olc 399	. . 3 |- ( ps -> ( -. ph \/ ps ) )
2		olc 399	. . 3 |- ( ch -> ( ph \/ ch ) )
3	1, 2	anim12i 590	. 2 |- ( ( ps /\ ch ) -> ( ( -. ph \/ ps ) /\ ( ph \/ ch ) ) )
4		dfifp4 1016	. 2 |- (if- ( ph , ps , ch ) <-> ( ( -. ph \/ ps ) /\ ( ph \/ ch ) ) )
5	3, 4	sylibr 224	1 |- ( ( ps /\ ch ) -> if- ( ph , ps , ch ) )

Syntax hints:   -. wn 3   -> wi 4   \/ 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:  bj-consensus  32562  bj-consensusALT  32563  axfrege58a  38168