Theorem ceqsexv

Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995) Avoid ax-12 . (Revised by Gino Giotto, 12-Oct-2024)

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

Proof

Step	Hyp	Ref	Expression
1		ceqsexv.1	|- A e. _V
2		ceqsexv.2	|- ( x = A -> ( ph <-> ps ) )
3	2	biimpa	|- ( ( x = A /\ ph ) -> ps )
4	3	exlimiv	|- ( E. x ( x = A /\ ph ) -> ps )
5	2	biimprcd	|- ( ps -> ( x = A -> ph ) )
6	5	alrimiv	|- ( ps -> A. x ( x = A -> ph ) )
7	1	isseti	|- E. x x = A
8		exintr	|- ( A. x ( x = A -> ph ) -> ( E. x x = A -> E. x ( x = A /\ ph ) ) )
9	6 7 8	mpisyl	|- ( ps -> E. x ( x = A /\ ph ) )
10	4 9	impbii	|- ( E. x ( x = A /\ ph ) <-> ps )