Theorem ifpbi12 37833

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

Assertion

Ref	Expression
ifpbi12	|- ( ( ( ph <-> ps ) /\ ( ch <-> th ) ) -> (if- ( ph , ch , ta ) <-> if- ( ps , th , ta ) ) )

Proof of Theorem ifpbi12

Step	Hyp	Ref	Expression
1		imbi12 336	. . . 4 |- ( ( ph <-> ps ) -> ( ( ch <-> th ) -> ( ( ph -> ch ) <-> ( ps -> th ) ) ) )
2	1	imp 445	. . 3 |- ( ( ( ph <-> ps ) /\ ( ch <-> th ) ) -> ( ( ph -> ch ) <-> ( ps -> th ) ) )
3		simpl 473	. . . . 5 |- ( ( ( ph <-> ps ) /\ ( ch <-> th ) ) -> ( ph <-> ps ) )
4	3	notbid 308	. . . 4 |- ( ( ( ph <-> ps ) /\ ( ch <-> th ) ) -> ( -. ph <-> -. ps ) )
5	4	imbi1d 331	. . 3 |- ( ( ( ph <-> ps ) /\ ( ch <-> th ) ) -> ( ( -. ph -> ta ) <-> ( -. ps -> ta ) ) )
6	2, 5	anbi12d 747	. 2 |- ( ( ( ph <-> ps ) /\ ( ch <-> th ) ) -> ( ( ( ph -> ch ) /\ ( -. ph -> ta ) ) <-> ( ( ps -> th ) /\ ( -. ps -> ta ) ) ) )
7		dfifp2 1014	. 2 |- (if- ( ph , ch , ta ) <-> ( ( ph -> ch ) /\ ( -. ph -> ta ) ) )
8		dfifp2 1014	. 2 |- (if- ( ps , th , ta ) <-> ( ( ps -> th ) /\ ( -. ps -> ta ) ) )
9	6, 7, 8	3bitr4g 303	1 |- ( ( ( ph <-> ps ) /\ ( ch <-> th ) ) -> (if- ( ph , ch , ta ) <-> if- ( ps , th , ta ) ) )

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)