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Theorem spcimgft 3435
 Description: A closed version of spcimgf 3437. (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 3364 . 2 (𝐴𝐵𝐴 ∈ V)
2 spcimgft.2 . . . . 5 𝑥𝐴
32issetf 3360 . . . 4 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
4 exim 1909 . . . 4 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝜑𝜓)))
53, 4syl5bi 232 . . 3 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴 ∈ V → ∃𝑥(𝜑𝜓)))
6 spcimgft.1 . . . 4 𝑥𝜓
7619.36 2254 . . 3 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
85, 7syl6ib 241 . 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 1629   = wceq 1631  ∃wex 1852  Ⅎwnf 1856   ∈ wcel 2145  Ⅎwnfc 2900  Vcvv 3351 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353 This theorem is referenced by:  spcgft  3436  spcimgf  3437  spcimdv  3441  ss2iundf  38477  spcdvw  42954
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