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Theorem elabf2 14394
Description: One implication of elabf 2880. (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 2319 . . 3 𝑥𝐴
3 elabf2.nf . . 3 𝑥𝜓
4 elabf2.1 . . 3 (𝑥 = 𝐴 → (𝜓𝜑))
52, 3, 4elabgf2 14392 . 2 (𝐴 ∈ V → (𝜓𝐴 ∈ {𝑥𝜑}))
61, 5ax-mp 5 1 (𝜓𝐴 ∈ {𝑥𝜑})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wnf 1460  wcel 2148  {cab 2163  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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739
This theorem is referenced by:  elab2a  14396  bj-bdfindis  14559
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