Theorem ifpbi3 37812

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

Assertion

Ref	Expression
ifpbi3	|- ( ( ph <-> ps ) -> (if- ( ch , th , ph ) <-> if- ( ch , th , ps ) ) )

Proof of Theorem ifpbi3

Step	Hyp	Ref	Expression
1		imbi2 338	. . 3 |- ( ( ph <-> ps ) -> ( ( -. ch -> ph ) <-> ( -. ch -> ps ) ) )
2	1	anbi2d 740	. 2 |- ( ( ph <-> ps ) -> ( ( ( ch -> th ) /\ ( -. ch -> ph ) ) <-> ( ( ch -> th ) /\ ( -. ch -> ps ) ) ) )
3		dfifp2 1014	. 2 |- (if- ( ch , th , ph ) <-> ( ( ch -> th ) /\ ( -. ch -> ph ) ) )
4		dfifp2 1014	. 2 |- (if- ( ch , th , ps ) <-> ( ( ch -> th ) /\ ( -. ch -> ps ) ) )
5	2, 3, 4	3bitr4g 303	1 |- ( ( ph <-> ps ) -> (if- ( ch , th , ph ) <-> if- ( ch , th , ps ) ) )

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:  ifpxorcor  37821  ifpnot23c  37829  ifpdfnan  37831