Theorem ifpbi123 37835

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

Assertion

Ref	Expression
ifpbi123	|- ( ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) -> (if- ( ph , ch , ta ) <-> if- ( ps , th , et ) ) )

Proof of Theorem ifpbi123

Step	Hyp	Ref	Expression
1		simp1 1061	. . . 4 |- ( ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) -> ( ph <-> ps ) )
2		simp2 1062	. . . 4 |- ( ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) -> ( ch <-> th ) )
3	1, 2	imbi12d 334	. . 3 |- ( ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) -> ( ( ph -> ch ) <-> ( ps -> th ) ) )
4	1	notbid 308	. . . 4 |- ( ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) -> ( -. ph <-> -. ps ) )
5		simp3 1063	. . . 4 |- ( ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) -> ( ta <-> et ) )
6	4, 5	imbi12d 334	. . 3 |- ( ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) -> ( ( -. ph -> ta ) <-> ( -. ps -> et ) ) )
7	3, 6	anbi12d 747	. 2 |- ( ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) -> ( ( ( ph -> ch ) /\ ( -. ph -> ta ) ) <-> ( ( ps -> th ) /\ ( -. ps -> et ) ) ) )
8		dfifp2 1014	. 2 |- (if- ( ph , ch , ta ) <-> ( ( ph -> ch ) /\ ( -. ph -> ta ) ) )
9		dfifp2 1014	. 2 |- (if- ( ps , th , et ) <-> ( ( ps -> th ) /\ ( -. ps -> et ) ) )
10	7, 8, 9	3bitr4g 303	1 |- ( ( ( ph <-> ps ) /\ ( ch <-> th ) /\ ( ta <-> et ) ) -> (if- ( ph , ch , ta ) <-> if- ( ps , th , et ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384  if-wif 1012   /\ w3a 1037

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  df-3an 1039

This theorem is referenced by: (None)