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Theorem spcimegft 2690
Description: A closed version of spcimegf 2693. (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 2624 . 2 (𝐴𝐵𝐴 ∈ V)
2 spcimgft.2 . . . . 5 𝑥𝐴
32issetf 2620 . . . 4 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
4 exim 1533 . . . 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 1607 . . 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 1285   = wceq 1287  wnf 1392  wex 1424  wcel 1436  wnfc 2212  Vcvv 2615
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617
This theorem is referenced by:  spcegft  2691  spcimegf  2693  spcimedv  2698
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