Theorem ifpbi1 37822

Description: Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.)

Assertion

Ref	Expression
ifpbi1	|- ( ( ph <-> ps ) -> (if- ( ph , ch , th ) <-> if- ( ps , ch , th ) ) )

Proof of Theorem ifpbi1

Step	Hyp	Ref	Expression
1		imbi1 337	. . 3 |- ( ( ph <-> ps ) -> ( ( ph -> ch ) <-> ( ps -> ch ) ) )
2		notbi 309	. . . . 5 |- ( ( ph <-> ps ) <-> ( -. ph <-> -. ps ) )
3	2	biimpi 206	. . . 4 |- ( ( ph <-> ps ) -> ( -. ph <-> -. ps ) )
4	3	imbi1d 331	. . 3 |- ( ( ph <-> ps ) -> ( ( -. ph -> th ) <-> ( -. ps -> th ) ) )
5	1, 4	anbi12d 747	. 2 |- ( ( ph <-> ps ) -> ( ( ( ph -> ch ) /\ ( -. ph -> th ) ) <-> ( ( ps -> ch ) /\ ( -. ps -> th ) ) ) )
6		dfifp2 1014	. 2 |- (if- ( ph , ch , th ) <-> ( ( ph -> ch ) /\ ( -. ph -> th ) ) )
7		dfifp2 1014	. 2 |- (if- ( ps , ch , th ) <-> ( ( ps -> ch ) /\ ( -. ps -> th ) ) )
8	5, 6, 7	3bitr4g 303	1 |- ( ( ph <-> ps ) -> (if- ( ph , ch , th ) <-> if- ( ps , ch , 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:  ifpimim  37854  axfrege52a  38150