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Theorem spcimgft 3578
Description: A closed version of spcimgf 3580. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
spcimgft.1 𝑥𝜓
spcimgft.2 𝑥𝐴
Assertion
Ref Expression
spcimgft (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝐵 → (∀𝑥𝜑𝜓)))

Proof of Theorem spcimgft
StepHypRef Expression
1 elex 3492 . 2 (𝐴𝐵𝐴 ∈ V)
2 spcimgft.2 . . . . 5 𝑥𝐴
32issetf 3488 . . . 4 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
4 exim 1835 . . . 4 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝜑𝜓)))
53, 4biimtrid 241 . . 3 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴 ∈ V → ∃𝑥(𝜑𝜓)))
6 spcimgft.1 . . . 4 𝑥𝜓
7619.36 2222 . . 3 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
85, 7imbitrdi 250 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴 ∈ V → (∀𝑥𝜑𝜓)))
91, 8syl5 34 1 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝐵 → (∀𝑥𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1538   = wceq 1540  wex 1780  wnf 1784  wcel 2105  wnfc 2882  Vcvv 3473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-v 3475
This theorem is referenced by:  spcgft  3579  spcimgf  3580  ss2iundf  42713  spcdvw  47813
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