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Theorem elabf2 9921
 Description: One implication of elabf 2686. (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 2178 . . 3 𝑥𝐴
3 elabf2.nf . . 3 𝑥𝜓
4 elabf2.1 . . 3 (𝑥 = 𝐴 → (𝜓𝜑))
52, 3, 4elabgf2 9919 . 2 (𝐴 ∈ V → (𝜓𝐴 ∈ {𝑥𝜑}))
61, 5ax-mp 7 1 (𝜓𝐴 ∈ {𝑥𝜑})
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243  Ⅎwnf 1349   ∈ wcel 1393  {cab 2026  Vcvv 2557 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559 This theorem is referenced by:  elab2a  9923  bj-bdfindis  10072
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