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Theorem elabf 2757
Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 12-Oct-2016.)
Hypotheses
Ref Expression
elabf.1 𝑥𝜓
elabf.2 𝐴 ∈ V
elabf.3 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
elabf (𝐴 ∈ {𝑥𝜑} ↔ 𝜓)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem elabf
StepHypRef Expression
1 elabf.2 . 2 𝐴 ∈ V
2 nfcv 2228 . . 3 𝑥𝐴
3 elabf.1 . . 3 𝑥𝜓
4 elabf.3 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
52, 3, 4elabgf 2756 . 2 (𝐴 ∈ V → (𝐴 ∈ {𝑥𝜑} ↔ 𝜓))
61, 5ax-mp 7 1 (𝐴 ∈ {𝑥𝜑} ↔ 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1289  wnf 1394  wcel 1438  {cab 2074  Vcvv 2619
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621
This theorem is referenced by:  elab  2758  indpi  6880
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