Theorem ifpbi12

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

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

Proof

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