Theorem spcimegf

Description: Existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017)

	Ref	Expression
Hypotheses	spcimgf.1	|- F/_ x A
	spcimgf.2	|- F/ x ps
	spcimegf.3	|- ( x = A -> ( ps -> ph ) )
Assertion	spcimegf	|- ( A e. V -> ( ps -> E. x ph ) )

Proof

Step	Hyp	Ref	Expression
1		spcimgf.1	|- F/_ x A
2		spcimgf.2	|- F/ x ps
3		spcimegf.3	|- ( x = A -> ( ps -> ph ) )
4	2	nfn	|- F/ x -. ps
5	3	con3d	|- ( x = A -> ( -. ph -> -. ps ) )
6	1 4 5	spcimgf	|- ( A e. V -> ( A. x -. ph -> -. ps ) )
7	6	con2d	|- ( A e. V -> ( ps -> -. A. x -. ph ) )
8		df-ex	|- ( E. x ph <-> -. A. x -. ph )
9	7 8	syl6ibr	|- ( A e. V -> ( ps -> E. x ph ) )