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Theorem ceqsexgv 2733
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 (𝐴𝑉 → (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem ceqsexgv
StepHypRef Expression
1 nfv 1462 . 2 𝑥𝜓
2 ceqsexgv.1 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
31, 2ceqsexg 2732 1 (𝐴𝑉 → (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1285  wex 1422  wcel 1434
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-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2613
This theorem is referenced by:  ceqsrexv  2734  clel3g  2738  elxp4  4859  elxp5  4860  dmfco  5295  fndmdif  5326  fndmin  5328  fmptco  5384  rexrnmpt2  5669  brtpos2  5922  xpsnen  6388  prarloc  6832
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