Theorem ifpbi2

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

	Ref	Expression
Assertion	ifpbi2	|- ( ( ph <-> ps ) -> ( if- ( ch , ph , th ) <-> if- ( ch , ps , th ) ) )

Proof

Step	Hyp	Ref	Expression
1		imbi2	|- ( ( ph <-> ps ) -> ( ( ch -> ph ) <-> ( ch -> ps ) ) )
2	1	anbi1d	|- ( ( ph <-> ps ) -> ( ( ( ch -> ph ) /\ ( -. ch -> th ) ) <-> ( ( ch -> ps ) /\ ( -. ch -> th ) ) ) )
3		dfifp2	|- ( if- ( ch , ph , th ) <-> ( ( ch -> ph ) /\ ( -. ch -> th ) ) )
4		dfifp2	|- ( if- ( ch , ps , th ) <-> ( ( ch -> ps ) /\ ( -. ch -> th ) ) )
5	2 3 4	3bitr4g	|- ( ( ph <-> ps ) -> ( if- ( ch , ph , th ) <-> if- ( ch , ps , th ) ) )