Theorem spcegf

Description: Existential specialization, using implicit substitution. (Contributed by NM, 2-Feb-1997)

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

Proof

Step	Hyp	Ref	Expression
1		spcgf.1	|- F/_ x A
2		spcgf.2	|- F/ x ps
3		spcgf.3	|- ( x = A -> ( ph <-> ps ) )
4	2	nfn	|- F/ x -. ps
5	3	notbid	|- ( x = A -> ( -. ph <-> -. ps ) )
6	1 4 5	spcgf	|- ( 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 ) )