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Theorem spcimgft 3525
Description: A closed version of spcimgf 3527. (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 3449 . 2 (𝐴𝐵𝐴 ∈ V)
2 spcimgft.2 . . . . 5 𝑥𝐴
32issetf 3445 . . . 4 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
4 exim 1840 . . . 4 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝜑𝜓)))
53, 4syl5bi 241 . . 3 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴 ∈ V → ∃𝑥(𝜑𝜓)))
6 spcimgft.1 . . . 4 𝑥𝜓
7619.36 2227 . . 3 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
85, 7syl6ib 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 1540   = wceq 1542  wex 1786  wnf 1790  wcel 2110  wnfc 2889  Vcvv 3431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-v 3433
This theorem is referenced by:  spcgft  3526  spcimgf  3527  ss2iundf  41237  spcdvw  46354
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