Theorem ifpid1g 37839

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

Assertion

Ref	Expression
ifpid1g	|- ( ( ph <-> if- ( ph , ps , ch ) ) <-> ( ( ch -> ph ) /\ ( ph -> ps ) ) )

Proof of Theorem ifpid1g

Step	Hyp	Ref	Expression
1		ifpidg 37836	. 2 |- ( ( ph <-> if- ( ph , ps , ch ) ) <-> ( ( ( ( ph /\ ps ) -> ph ) /\ ( ( ph /\ ph ) -> ps ) ) /\ ( ( ch -> ( ph \/ ph ) ) /\ ( ph -> ( ph \/ ch ) ) ) ) )
2		ancom 466	. 2 |- ( ( ( ( ( ph /\ ps ) -> ph ) /\ ( ( ph /\ ph ) -> ps ) ) /\ ( ( ch -> ( ph \/ ph ) ) /\ ( ph -> ( ph \/ ch ) ) ) ) <-> ( ( ( ch -> ( ph \/ ph ) ) /\ ( ph -> ( ph \/ ch ) ) ) /\ ( ( ( ph /\ ps ) -> ph ) /\ ( ( ph /\ ph ) -> ps ) ) ) )
3		pm4.25 537	. . . . 5 |- ( ph <-> ( ph \/ ph ) )
4	3	imbi2i 326	. . . 4 |- ( ( ch -> ph ) <-> ( ch -> ( ph \/ ph ) ) )
5		orc 400	. . . . 5 |- ( ph -> ( ph \/ ch ) )
6	5	biantru 526	. . . 4 |- ( ( ch -> ( ph \/ ph ) ) <-> ( ( ch -> ( ph \/ ph ) ) /\ ( ph -> ( ph \/ ch ) ) ) )
7	4, 6	bitr2i 265	. . 3 |- ( ( ( ch -> ( ph \/ ph ) ) /\ ( ph -> ( ph \/ ch ) ) ) <-> ( ch -> ph ) )
8		pm4.24 675	. . . . 5 |- ( ph <-> ( ph /\ ph ) )
9	8	imbi1i 339	. . . 4 |- ( ( ph -> ps ) <-> ( ( ph /\ ph ) -> ps ) )
10		simpl 473	. . . . 5 |- ( ( ph /\ ps ) -> ph )
11	10	biantrur 527	. . . 4 |- ( ( ( ph /\ ph ) -> ps ) <-> ( ( ( ph /\ ps ) -> ph ) /\ ( ( ph /\ ph ) -> ps ) ) )
12	9, 11	bitr2i 265	. . 3 |- ( ( ( ( ph /\ ps ) -> ph ) /\ ( ( ph /\ ph ) -> ps ) ) <-> ( ph -> ps ) )
13	7, 12	anbi12i 733	. 2 |- ( ( ( ( ch -> ( ph \/ ph ) ) /\ ( ph -> ( ph \/ ch ) ) ) /\ ( ( ( ph /\ ps ) -> ph ) /\ ( ( ph /\ ph ) -> ps ) ) ) <-> ( ( ch -> ph ) /\ ( ph -> ps ) ) )
14	1, 2, 13	3bitri 286	1 |- ( ( ph <-> if- ( ph , ps , ch ) ) <-> ( ( ch -> ph ) /\ ( 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)