Theorem ifpbi3

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

	Ref	Expression
Assertion	ifpbi3	|- ( ( ph <-> ps ) -> ( if- ( ch , th , ph ) <-> if- ( ch , th , ps ) ) )

Proof

Step	Hyp	Ref	Expression
1		imbi2	|- ( ( ph <-> ps ) -> ( ( -. ch -> ph ) <-> ( -. ch -> ps ) ) )
2	1	anbi2d	|- ( ( ph <-> ps ) -> ( ( ( ch -> th ) /\ ( -. ch -> ph ) ) <-> ( ( ch -> th ) /\ ( -. ch -> ps ) ) ) )
3		dfifp2	|- ( if- ( ch , th , ph ) <-> ( ( ch -> th ) /\ ( -. ch -> ph ) ) )
4		dfifp2	|- ( if- ( ch , th , ps ) <-> ( ( ch -> th ) /\ ( -. ch -> ps ) ) )
5	2 3 4	3bitr4g	|- ( ( ph <-> ps ) -> ( if- ( ch , th , ph ) <-> if- ( ch , th , ps ) ) )