Theorem r19.29af2

Description: A commonly used pattern based on r19.29 . (Contributed by Thierry Arnoux, 17-Dec-2017) (Proof shortened by OpenAI, 25-Mar-2020)

	Ref	Expression
Hypotheses	r19.29af2.p	|- F/ x ph
	r19.29af2.c	|- F/ x ch
	r19.29af2.1	|- ( ( ( ph /\ x e. A ) /\ ps ) -> ch )
	r19.29af2.2	|- ( ph -> E. x e. A ps )
Assertion	r19.29af2	|- ( ph -> ch )

Proof

Step	Hyp	Ref	Expression
1		r19.29af2.p	|- F/ x ph
2		r19.29af2.c	|- F/ x ch
3		r19.29af2.1	|- ( ( ( ph /\ x e. A ) /\ ps ) -> ch )
4		r19.29af2.2	|- ( ph -> E. x e. A ps )
5	3	exp31	|- ( ph -> ( x e. A -> ( ps -> ch ) ) )
6	1 2 5	rexlimd	|- ( ph -> ( E. x e. A ps -> ch ) )
7	4 6	mpd	|- ( ph -> ch )