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Theorem ceqsexgv 2667
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 1418 . 2 xψ
2 ceqsexgv.1 . 2 (x = A → (φψ))
31, 2ceqsexg 2666 1 (A 𝑉 → (x(x = A φ) ↔ ψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wex 1378   wcel 1390
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553
This theorem is referenced by:  ceqsrexv  2668  clel3g  2672  elxp4  4751  elxp5  4752  dmfco  5184  fndmdif  5215  fndmin  5217  fmptco  5273  rexrnmpt2  5558  brtpos2  5807  xpsnen  6231  prarloc  6485
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