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Theorem ceqsex 3145
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
ceqsex.2
ceqsex.3
Assertion
Ref Expression
ceqsex
Distinct variable group:   ,

Proof of Theorem ceqsex
StepHypRef Expression
1 ceqsex.1 . . 3
2 ceqsex.3 . . . 4
32biimpa 484 . . 3
41, 3exlimi 1912 . 2
52biimprcd 225 . . . 4
61, 5alrimi 1877 . . 3
7 ceqsex.2 . . . 4
87isseti 3115 . . 3
9 exintr 1702 . . 3
106, 8, 9mpisyl 18 . 2
114, 10impbii 188 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  =wceq 1395  E.wex 1612  F/wnf 1616  e.wcel 1818   cvv 3109
This theorem is referenced by:  ceqsexv  3146  ceqsex2  3147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-12 1854  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-v 3111
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