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Theorem elabgf1 13775
Description: One implication of elabgf 2872. (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 13774 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴 ∈ {𝑥𝜑} → 𝜓))
4 elabgf1.1 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
53, 4mpg 1444 1 (𝐴 ∈ {𝑥𝜑} → 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1348  wnf 1453  wcel 2141  {cab 2156  wnfc 2299
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732
This theorem is referenced by:  elabf1  13777
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