Theorem ifpbi2 37811

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

Assertion

Ref	Expression
ifpbi2	|- ( ( ph <-> ps ) -> (if- ( ch , ph , th ) <-> if- ( ch , ps , th ) ) )

Proof of Theorem ifpbi2

Step	Hyp	Ref	Expression
1		imbi2 338	. . 3 |- ( ( ph <-> ps ) -> ( ( ch -> ph ) <-> ( ch -> ps ) ) )
2	1	anbi1d 741	. 2 |- ( ( ph <-> ps ) -> ( ( ( ch -> ph ) /\ ( -. ch -> th ) ) <-> ( ( ch -> ps ) /\ ( -. ch -> th ) ) ) )
3		dfifp2 1014	. 2 |- (if- ( ch , ph , th ) <-> ( ( ch -> ph ) /\ ( -. ch -> th ) ) )
4		dfifp2 1014	. 2 |- (if- ( ch , ps , th ) <-> ( ( ch -> ps ) /\ ( -. ch -> th ) ) )
5	2, 3, 4	3bitr4g 303	1 |- ( ( ph <-> ps ) -> (if- ( ch , ph , th ) <-> if- ( ch , 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:  ifpnot23b  37827