Theorem spcimedv

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

	Ref	Expression
Hypotheses	spcimdv.1	|- ( ph -> A e. B )
	spcimedv.2	|- ( ( ph /\ x = A ) -> ( ch -> ps ) )
Assertion	spcimedv	|- ( ph -> ( ch -> E. x ps ) )

Proof

Step	Hyp	Ref	Expression
1		spcimdv.1	|- ( ph -> A e. B )
2		spcimedv.2	|- ( ( ph /\ x = A ) -> ( ch -> ps ) )
3	2	con3d	|- ( ( ph /\ x = A ) -> ( -. ps -> -. ch ) )
4	1 3	spcimdv	|- ( ph -> ( A. x -. ps -> -. ch ) )
5	4	con2d	|- ( ph -> ( ch -> -. A. x -. ps ) )
6		df-ex	|- ( E. x ps <-> -. A. x -. ps )
7	5 6	syl6ibr	|- ( ph -> ( ch -> E. x ps ) )