Theorem ceqsexv2d 2813

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.)

Hypotheses

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

Assertion

Ref	Expression
ceqsexv2d	|- E. x ph

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   ps( x)

Proof of Theorem ceqsexv2d

Step	Hyp	Ref	Expression
1		ceqsexv2d.1	. . 3 |- A e. _V
2	1	isseti 2781	. 2 |- E. x x = A
3		ceqsexv2d.3	. . 3 |- ps
4		ceqsexv2d.2	. . 3 |- ( x = A -> ( ph <-> ps ) )
5	3, 4	mpbiri 168	. 2 |- ( x = A -> ph )
6	2, 5	eximii 1626	1 |- E. x ph

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   E.wex 1516   e. wcel 2177   _Vcvv 2773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-v 2775

This theorem is referenced by: (None)