Theorem anifp

Description: The conditional operator is implied by the conjunction of its possible outputs. Dual statement of ifpor . (Contributed by BJ, 30-Sep-2019)

	Ref	Expression
Assertion	anifp	|- ( ( ps /\ ch ) -> if- ( ph , ps , ch ) )

Proof

Step	Hyp	Ref	Expression
1		olc	|- ( ps -> ( -. ph \/ ps ) )
2		olc	|- ( ch -> ( ph \/ ch ) )
3	1 2	anim12i	|- ( ( ps /\ ch ) -> ( ( -. ph \/ ps ) /\ ( ph \/ ch ) ) )
4		dfifp4	|- ( if- ( ph , ps , ch ) <-> ( ( -. ph \/ ps ) /\ ( ph \/ ch ) ) )
5	3 4	sylibr	|- ( ( ps /\ ch ) -> if- ( ph , ps , ch ) )