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Theorem spcegft 2739
Description: A closed version of spcegf 2743. (Contributed by Jim Kingdon, 22-Jun-2018.)
Hypotheses
Ref Expression
spcimgft.1 𝑥𝜓
spcimgft.2 𝑥𝐴
Assertion
Ref Expression
spcegft (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝐵 → (𝜓 → ∃𝑥𝜑)))

Proof of Theorem spcegft
StepHypRef Expression
1 bi2 129 . . . 4 ((𝜑𝜓) → (𝜓𝜑))
21imim2i 12 . . 3 ((𝑥 = 𝐴 → (𝜑𝜓)) → (𝑥 = 𝐴 → (𝜓𝜑)))
32alimi 1416 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → ∀𝑥(𝑥 = 𝐴 → (𝜓𝜑)))
4 spcimgft.1 . . 3 𝑥𝜓
5 spcimgft.2 . . 3 𝑥𝐴
64, 5spcimegft 2738 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜓𝜑)) → (𝐴𝐵 → (𝜓 → ∃𝑥𝜑)))
73, 6syl 14 1 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝐵 → (𝜓 → ∃𝑥𝜑)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1314   = wceq 1316  wnf 1421  wex 1453  wcel 1465  wnfc 2245
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662
This theorem is referenced by:  spcegf  2743
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