Theorem ceqsex 2815

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 296	. . 3 |- ( ( x = A /\ ph ) -> ps )
4	1, 3	exlimi 1618	. 2 |- ( E. x ( x = A /\ ph ) -> ps )
5	2	biimprcd 160	. . . 4 |- ( ps -> ( x = A -> ph ) )
6	1, 5	alrimi 1546	. . 3 |- ( ps -> A. x ( x = A -> ph ) )
7		ceqsex.2	. . . 4 |- A e. _V
8	7	isseti 2785	. . 3 |- E. x x = A
9		exintr 1658	. . 3 |- ( A. x ( x = A -> ph ) -> ( E. x x = A -> E. x ( x = A /\ ph ) ) )
10	6, 8, 9	mpisyl 1467	. 2 |- ( ps -> E. x ( x = A /\ ph ) )
11	4, 10	impbii 126	1 |- ( E. x ( x = A /\ ph ) <-> ps )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1371   = wceq 1373   F/wnf 1484   E.wex 1516   e. wcel 2178   _Vcvv 2776

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 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-v 2778

This theorem is referenced by:  ceqsexv  2816  ceqsex2  2818