Theorem ceqsex 2657

Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) (Revised by Mario Carneiro, 10-Oct-2016.)

Hypotheses

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

Assertion

Ref	Expression
ceqsex	|- ( E. x ( x = A /\ ph ) <-> ps )

Distinct variable group:   x, A

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

Proof of Theorem ceqsex

Step	Hyp	Ref	Expression
1		ceqsex.1	. . 3 |- F/ x ps
2		ceqsex.3	. . . 4 |- ( x = A -> ( ph <-> ps ) )
3	2	biimpa 290	. . 3 |- ( ( x = A /\ ph ) -> ps )
4	1, 3	exlimi 1530	. 2 |- ( E. x ( x = A /\ ph ) -> ps )
5	2	biimprcd 158	. . . 4 |- ( ps -> ( x = A -> ph ) )
6	1, 5	alrimi 1460	. . 3 |- ( ps -> A. x ( x = A -> ph ) )
7		ceqsex.2	. . . 4 |- A e. _V
8	7	isseti 2627	. . 3 |- E. x x = A
9		exintr 1570	. . 3 |- ( A. x ( x = A -> ph ) -> ( E. x x = A -> E. x ( x = A /\ ph ) ) )
10	6, 8, 9	mpisyl 1380	. 2 |- ( ps -> E. x ( x = A /\ ph ) )
11	4, 10	impbii 124	1 |- ( E. x ( x = A /\ ph ) <-> ps )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1287   = wceq 1289   F/wnf 1394   E.wex 1426   e. wcel 1438   _Vcvv 2619

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621

This theorem is referenced by:  ceqsexv  2658  ceqsex2  2659