Theorem ceqsexv2d 3243

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

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 groups:   x, A   ps, x

Allowed substitution hint:   ph( x)

Proof of Theorem ceqsexv2d

Step	Hyp	Ref	Expression
1		ceqsexv2d.3	. 2 |- ps
2		ceqsexv2d.1	. . . 4 |- A e. _V
3		ceqsexv2d.2	. . . 4 |- ( x = A -> ( ph <-> ps ) )
4	2, 3	ceqsexv 3242	. . 3 |- ( E. x ( x = A /\ ph ) <-> ps )
5	4	biimpri 218	. 2 |- ( ps -> E. x ( x = A /\ ph ) )
6		exsimpr 1796	. 2 |- ( E. x ( x = A /\ ph ) -> E. x ph )
7	1, 5, 6	mp2b 10	1 |- E. x ph

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202

This theorem is referenced by:  2lgslem1  25119  griedg0prc  26156  1loopgrvd2  26399