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Theorem ceqsexgv 2650
Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 29-Dec-1996.)
Hypothesis
Ref Expression
ceqsexgv.1 (x = A → (φψ))
Assertion
Ref Expression
ceqsexgv (A 𝑉 → (x(x = A φ) ↔ ψ))
Distinct variable groups:   x,A   ψ,x
Allowed substitution hints:   φ(x)   𝑉(x)

Proof of Theorem ceqsexgv
StepHypRef Expression
1 nfv 1402 . 2 xψ
2 ceqsexgv.1 . 2 (x = A → (φψ))
31, 2ceqsexg 2649 1 (A 𝑉 → (x(x = A φ) ↔ ψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228  wex 1362   wcel 1374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537
This theorem is referenced by:  ceqsrexv  2651  clel3g  2655  elxp4  4735  elxp5  4736  dmfco  5166  fndmdif  5197  fndmin  5199  fmptco  5255  rexrnmpt2  5539  brtpos2  5788  prarloc  6357
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