Theorem ifpbi1

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

	Ref	Expression
Assertion	ifpbi1	|- ( ( ph <-> ps ) -> ( if- ( ph , ch , th ) <-> if- ( ps , ch , th ) ) )

Proof

Step	Hyp	Ref	Expression
1		imbi1	|- ( ( ph <-> ps ) -> ( ( ph -> ch ) <-> ( ps -> ch ) ) )
2		notbi	|- ( ( ph <-> ps ) <-> ( -. ph <-> -. ps ) )
3	2	biimpi	|- ( ( ph <-> ps ) -> ( -. ph <-> -. ps ) )
4	3	imbi1d	|- ( ( ph <-> ps ) -> ( ( -. ph -> th ) <-> ( -. ps -> th ) ) )
5	1 4	anbi12d	|- ( ( ph <-> ps ) -> ( ( ( ph -> ch ) /\ ( -. ph -> th ) ) <-> ( ( ps -> ch ) /\ ( -. ps -> th ) ) ) )
6		dfifp2	|- ( if- ( ph , ch , th ) <-> ( ( ph -> ch ) /\ ( -. ph -> th ) ) )
7		dfifp2	|- ( if- ( ps , ch , th ) <-> ( ( ps -> ch ) /\ ( -. ps -> th ) ) )
8	5 6 7	3bitr4g	|- ( ( ph <-> ps ) -> ( if- ( ph , ch , th ) <-> if- ( ps , ch , th ) ) )