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Theorem spcimegft 2685
Description: A closed version of spcimegf 2688. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
spcimgft.1 𝑥𝜓
spcimgft.2 𝑥𝐴
Assertion
Ref Expression
spcimegft (∀𝑥(𝑥 = 𝐴 → (𝜓𝜑)) → (𝐴𝐵 → (𝜓 → ∃𝑥𝜑)))

Proof of Theorem spcimegft
StepHypRef Expression
1 elex 2619 . 2 (𝐴𝐵𝐴 ∈ V)
2 spcimgft.2 . . . . 5 𝑥𝐴
32issetf 2615 . . . 4 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
4 exim 1531 . . . 4 (∀𝑥(𝑥 = 𝐴 → (𝜓𝜑)) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝜓𝜑)))
53, 4syl5bi 150 . . 3 (∀𝑥(𝑥 = 𝐴 → (𝜓𝜑)) → (𝐴 ∈ V → ∃𝑥(𝜓𝜑)))
6 spcimgft.1 . . . 4 𝑥𝜓
7619.37-1 1605 . . 3 (∃𝑥(𝜓𝜑) → (𝜓 → ∃𝑥𝜑))
85, 7syl6 33 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜓𝜑)) → (𝐴 ∈ V → (𝜓 → ∃𝑥𝜑)))
91, 8syl5 32 1 (∀𝑥(𝑥 = 𝐴 → (𝜓𝜑)) → (𝐴𝐵 → (𝜓 → ∃𝑥𝜑)))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1283   = wceq 1285  wnf 1390  wex 1422  wcel 1434  wnfc 2210  Vcvv 2610
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612
This theorem is referenced by:  spcegft  2686  spcimegf  2688  spcimedv  2693
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