Theorem ceqsexv2d

Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by Thierry Arnoux, 10-Sep-2016)

	Ref	Expression
Hypotheses	ceqsexv2d.1	|- A e. _V
	ceqsexv2d.2	|- ( x = A -> ( ph <-> ps ) )
	ceqsexv2d.3	|- ps
Assertion	ceqsexv2d	|- E. x ph

Proof

Step	Hyp	Ref	Expression
1		ceqsexv2d.1	|- A e. _V
2		ceqsexv2d.2	|- ( x = A -> ( ph <-> ps ) )
3		ceqsexv2d.3	|- ps
4	1 2	ceqsexv	|- ( E. x ( x = A /\ ph ) <-> ps )
5	4	biimpri	|- ( ps -> E. x ( x = A /\ ph ) )
6		exsimpr	|- ( E. x ( x = A /\ ph ) -> E. x ph )
7	3 5 6	mp2b	|- E. x ph