Theorem spc2ev

Description: Existential specialization, using implicit substitution. (Contributed by NM, 3-Aug-1995)

	Ref	Expression
Hypotheses	spc2ev.1	|- A e. _V
	spc2ev.2	|- B e. _V
	spc2ev.3	|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
Assertion	spc2ev	|- ( ps -> E. x E. y ph )

Proof

Step	Hyp	Ref	Expression
1		spc2ev.1	|- A e. _V
2		spc2ev.2	|- B e. _V
3		spc2ev.3	|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
4	3	spc2egv	|- ( ( A e. _V /\ B e. _V ) -> ( ps -> E. x E. y ph ) )
5	1 2 4	mp2an	|- ( ps -> E. x E. y ph )