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Theorem ceqsexv 2776
Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.)
Hypotheses
Ref Expression
ceqsexv.1 𝐴 ∈ V
ceqsexv.2 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
ceqsexv (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ceqsexv
StepHypRef Expression
1 nfv 1528 . 2 𝑥𝜓
2 ceqsexv.1 . 2 𝐴 ∈ V
3 ceqsexv.2 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
41, 2, 3ceqsex 2775 1 (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wex 1492  wcel 2148  Vcvv 2737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-v 2739
This theorem is referenced by:  ceqsex3v  2779  gencbvex  2783  sbhypf  2786  euxfr2dc  2922  inuni  4152  eqvinop  4240  onm  4398  uniuni  4448  opeliunxp  4678  elvvv  4686  rexiunxp  4765  imai  4980  coi1  5140  abrexco  5754  opabex3d  6116  opabex3  6117  mapsnen  6805  xpsnen  6815  xpcomco  6820  xpassen  6824
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