Theorem ifpbi23 37817

Description: Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)

Assertion

Ref	Expression
ifpbi23	|- ( ( ( ph <-> ps ) /\ ( ch <-> th ) ) -> (if- ( ta , ph , ch ) <-> if- ( ta , ps , th ) ) )

Proof of Theorem ifpbi23

Step	Hyp	Ref	Expression
1		simpl 473	. . . 4 |- ( ( ( ph <-> ps ) /\ ( ch <-> th ) ) -> ( ph <-> ps ) )
2	1	imbi2d 330	. . 3 |- ( ( ( ph <-> ps ) /\ ( ch <-> th ) ) -> ( ( ta -> ph ) <-> ( ta -> ps ) ) )
3		simpr 477	. . . 4 |- ( ( ( ph <-> ps ) /\ ( ch <-> th ) ) -> ( ch <-> th ) )
4	3	imbi2d 330	. . 3 |- ( ( ( ph <-> ps ) /\ ( ch <-> th ) ) -> ( ( -. ta -> ch ) <-> ( -. ta -> th ) ) )
5	2, 4	anbi12d 747	. 2 |- ( ( ( ph <-> ps ) /\ ( ch <-> th ) ) -> ( ( ( ta -> ph ) /\ ( -. ta -> ch ) ) <-> ( ( ta -> ps ) /\ ( -. ta -> th ) ) ) )
6		dfifp2 1014	. 2 |- (if- ( ta , ph , ch ) <-> ( ( ta -> ph ) /\ ( -. ta -> ch ) ) )
7		dfifp2 1014	. 2 |- (if- ( ta , ps , th ) <-> ( ( ta -> ps ) /\ ( -. ta -> th ) ) )
8	5, 6, 7	3bitr4g 303	1 |- ( ( ( ph <-> ps ) /\ ( ch <-> th ) ) -> (if- ( ta , ph , ch ) <-> if- ( ta , ps , th ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ 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:  ifpdfbi  37818  ifpnot23d  37830  ifpdfxor  37832  ifpananb  37851  ifpnannanb  37852  ifpxorxorb  37856