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Theorem elabgf1 14758
Description: One implication of elabgf 2891. (Contributed by BJ, 21-Nov-2019.)
Hypotheses
Ref Expression
elabgf1.nf1 𝑥𝐴
elabgf1.nf2 𝑥𝜓
elabgf1.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
elabgf1 (𝐴 ∈ {𝑥𝜑} → 𝜓)

Proof of Theorem elabgf1
StepHypRef Expression
1 elabgf1.nf1 . . 3 𝑥𝐴
2 elabgf1.nf2 . . 3 𝑥𝜓
31, 2elabgft1 14757 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴 ∈ {𝑥𝜑} → 𝜓))
4 elabgf1.1 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
53, 4mpg 1461 1 (𝐴 ∈ {𝑥𝜑} → 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1363  wnf 1470  wcel 2158  {cab 2173  wnfc 2316
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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751
This theorem is referenced by:  elabf1  14760
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