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Theorem elabgf1 12986
 Description: One implication of elabgf 2826. (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 12985 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴 ∈ {𝑥𝜑} → 𝜓))
4 elabgf1.1 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
53, 4mpg 1427 1 (𝐴 ∈ {𝑥𝜑} → 𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1331  Ⅎwnf 1436   ∈ wcel 1480  {cab 2125  Ⅎwnfc 2268 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688 This theorem is referenced by:  elabf1  12988
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