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Theorem spcegft 2632
Description: A closed version of spcegf 2636. (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 121 . . . 4 ((𝜑𝜓) → (𝜓𝜑))
21imim2i 12 . . 3 ((𝑥 = 𝐴 → (𝜑𝜓)) → (𝑥 = 𝐴 → (𝜓𝜑)))
32alimi 1344 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → ∀𝑥(𝑥 = 𝐴 → (𝜓𝜑)))
4 spcimgft.1 . . 3 𝑥𝜓
5 spcimgft.2 . . 3 𝑥𝐴
64, 5spcimegft 2631 . 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 98  wal 1241   = wceq 1243  wnf 1349  wex 1381  wcel 1393  wnfc 2165
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:  spcegf  2636
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