Theorem ceqsexv2d

Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by Thierry Arnoux, 10-Sep-2016) Shorten, reduce dv conditions. (Revised by Wolf Lammen, 5-Jun-2025) (Proof shortened by SN, 5-Jun-2025)

	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	isseti	|- E. x x = A
5	3 2	mpbiri	|- ( x = A -> ph )
6	4 5	eximii	|- E. x ph