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Theorem elabf2 12916
Description: One implication of elabf 2801. (Contributed by BJ, 21-Nov-2019.)
Hypotheses
Ref Expression
elabf2.nf 𝑥𝜓
elabf2.s 𝐴 ∈ V
elabf2.1 (𝑥 = 𝐴 → (𝜓𝜑))
Assertion
Ref Expression
elabf2 (𝜓𝐴 ∈ {𝑥𝜑})
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem elabf2
StepHypRef Expression
1 elabf2.s . 2 𝐴 ∈ V
2 nfcv 2258 . . 3 𝑥𝐴
3 elabf2.nf . . 3 𝑥𝜓
4 elabf2.1 . . 3 (𝑥 = 𝐴 → (𝜓𝜑))
52, 3, 4elabgf2 12914 . 2 (𝐴 ∈ V → (𝜓𝐴 ∈ {𝑥𝜑}))
61, 5ax-mp 5 1 (𝜓𝐴 ∈ {𝑥𝜑})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1316  wnf 1421  wcel 1465  {cab 2103  Vcvv 2660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662
This theorem is referenced by:  elab2a  12918  bj-bdfindis  13072
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