Theorem ifpbi13 37834

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

Assertion

Ref	Expression
ifpbi13	|- ( ( ( ph <-> ps ) /\ ( ch <-> th ) ) -> (if- ( ph , ta , ch ) <-> if- ( ps , ta , th ) ) )

Proof of Theorem ifpbi13

Step	Hyp	Ref	Expression
1		simpl 473	. . . 4 |- ( ( ( ph <-> ps ) /\ ( ch <-> th ) ) -> ( ph <-> ps ) )
2	1	imbi1d 331	. . 3 |- ( ( ( ph <-> ps ) /\ ( ch <-> th ) ) -> ( ( ph -> ta ) <-> ( ps -> ta ) ) )
3		notbi 309	. . . . 5 |- ( ( ph <-> ps ) <-> ( -. ph <-> -. ps ) )
4		imbi12 336	. . . . 5 |- ( ( -. ph <-> -. ps ) -> ( ( ch <-> th ) -> ( ( -. ph -> ch ) <-> ( -. ps -> th ) ) ) )
5	3, 4	sylbi 207	. . . 4 |- ( ( ph <-> ps ) -> ( ( ch <-> th ) -> ( ( -. ph -> ch ) <-> ( -. ps -> th ) ) ) )
6	5	imp 445	. . 3 |- ( ( ( ph <-> ps ) /\ ( ch <-> th ) ) -> ( ( -. ph -> ch ) <-> ( -. ps -> th ) ) )
7	2, 6	anbi12d 747	. 2 |- ( ( ( ph <-> ps ) /\ ( ch <-> th ) ) -> ( ( ( ph -> ta ) /\ ( -. ph -> ch ) ) <-> ( ( ps -> ta ) /\ ( -. ps -> th ) ) ) )
8		dfifp2 1014	. 2 |- (if- ( ph , ta , ch ) <-> ( ( ph -> ta ) /\ ( -. ph -> ch ) ) )
9		dfifp2 1014	. 2 |- (if- ( ps , ta , th ) <-> ( ( ps -> ta ) /\ ( -. ps -> th ) ) )
10	7, 8, 9	3bitr4g 303	1 |- ( ( ( ph <-> ps ) /\ ( ch <-> th ) ) -> (if- ( ph , ta , ch ) <-> if- ( ps , ta , 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: (None)