Theorem ceqsex 3241

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 501	. . 3 |- ( ( x = A /\ ph ) -> ps )
4	1, 3	exlimi 2086	. 2 |- ( E. x ( x = A /\ ph ) -> ps )
5	2	biimprcd 240	. . . 4 |- ( ps -> ( x = A -> ph ) )
6	1, 5	alrimi 2082	. . 3 |- ( ps -> A. x ( x = A -> ph ) )
7		ceqsex.2	. . . 4 |- A e. _V
8	7	isseti 3209	. . 3 |- E. x x = A
9		exintr 1819	. . 3 |- ( A. x ( x = A -> ph ) -> ( E. x x = A -> E. x ( x = A /\ ph ) ) )
10	6, 8, 9	mpisyl 21	. 2 |- ( ps -> E. x ( x = A /\ ph ) )
11	4, 10	impbii 199	1 |- ( E. x ( x = A /\ ph ) <-> ps )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   F/wnf 1708   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:  ceqsexv  3242  ceqsex2  3244