Theorem r19.29af

Description: A commonly used pattern based on r19.29 . See r19.29a , r19.29an for a variant when x is disjoint from ph . (Contributed by Thierry Arnoux, 29-Nov-2017)

	Ref	Expression
Hypotheses	r19.29af.0	|- F/ x ph
	r19.29af.1	|- ( ( ( ph /\ x e. A ) /\ ps ) -> ch )
	r19.29af.2	|- ( ph -> E. x e. A ps )
Assertion	r19.29af	|- ( ph -> ch )

Proof

Step	Hyp	Ref	Expression
1		r19.29af.0	|- F/ x ph
2		r19.29af.1	|- ( ( ( ph /\ x e. A ) /\ ps ) -> ch )
3		r19.29af.2	|- ( ph -> E. x e. A ps )
4		nfv	|- F/ x ch
5	1 4 2 3	r19.29af2	|- ( ph -> ch )