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Theorem spcimgft 3424
Description: A closed version of spcimgf 3426. (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 3352 . 2 (𝐴𝐵𝐴 ∈ V)
2 spcimgft.2 . . . . 5 𝑥𝐴
32issetf 3348 . . . 4 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
4 exim 1910 . . . 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 2245 . . 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 1630   = wceq 1632  wex 1853  wnf 1857  wcel 2139  wnfc 2889  Vcvv 3340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342
This theorem is referenced by:  spcgft  3425  spcimgf  3426  spcimdv  3430  ss2iundf  38453  spcdvw  42936
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