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Theorem spcimgft 3499
Description: A closed version of spcimgf 3501. (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 3427 . 2 (𝐴𝐵𝐴 ∈ V)
2 spcimgft.2 . . . . 5 𝑥𝐴
32issetf 3422 . . . 4 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
4 exim 1796 . . . 4 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝜑𝜓)))
53, 4syl5bi 234 . . 3 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴 ∈ V → ∃𝑥(𝜑𝜓)))
6 spcimgft.1 . . . 4 𝑥𝜓
7619.36 2162 . . 3 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
85, 7syl6ib 243 . 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 1505   = wceq 1507  wex 1742  wnf 1746  wcel 2050  wnfc 2910  Vcvv 3409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2744
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-v 3411
This theorem is referenced by:  spcgft  3500  spcimgf  3501  ss2iundf  39367  spcdvw  44149
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