Theorem ifpid2g 37838

Description: Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)

Assertion

Ref	Expression
ifpid2g	|- ( ( ps <-> if- ( ph , ps , ch ) ) <-> ( ( ps -> ( ph \/ ch ) ) /\ ( ch -> ( ph \/ ps ) ) ) )

Proof of Theorem ifpid2g

Step	Hyp	Ref	Expression
1		ifpidg 37836	. 2 |- ( ( ps <-> if- ( ph , ps , ch ) ) <-> ( ( ( ( ph /\ ps ) -> ps ) /\ ( ( ph /\ ps ) -> ps ) ) /\ ( ( ch -> ( ph \/ ps ) ) /\ ( ps -> ( ph \/ ch ) ) ) ) )
2		simpr 477	. . . 4 |- ( ( ph /\ ps ) -> ps )
3	2, 2	pm3.2i 471	. . 3 |- ( ( ( ph /\ ps ) -> ps ) /\ ( ( ph /\ ps ) -> ps ) )
4	3	biantrur 527	. 2 |- ( ( ( ch -> ( ph \/ ps ) ) /\ ( ps -> ( ph \/ ch ) ) ) <-> ( ( ( ( ph /\ ps ) -> ps ) /\ ( ( ph /\ ps ) -> ps ) ) /\ ( ( ch -> ( ph \/ ps ) ) /\ ( ps -> ( ph \/ ch ) ) ) ) )
5		ancom 466	. 2 |- ( ( ( ch -> ( ph \/ ps ) ) /\ ( ps -> ( ph \/ ch ) ) ) <-> ( ( ps -> ( ph \/ ch ) ) /\ ( ch -> ( ph \/ ps ) ) ) )
6	1, 4, 5	3bitr2i 288	1 |- ( ( ps <-> if- ( ph , ps , ch ) ) <-> ( ( ps -> ( ph \/ ch ) ) /\ ( ch -> ( ph \/ ps ) ) ) )

Syntax hints:   -> wi 4   <-> 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)