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Theorem spcegf 3258
Description: Existential specialization, using implicit substitution. (Contributed by NM, 2-Feb-1997.)
Hypotheses
Ref Expression
spcgf.1 𝑥𝐴
spcgf.2 𝑥𝜓
spcgf.3 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
spcegf (𝐴𝑉 → (𝜓 → ∃𝑥𝜑))

Proof of Theorem spcegf
StepHypRef Expression
1 spcgf.1 . . . 4 𝑥𝐴
2 spcgf.2 . . . . 5 𝑥𝜓
32nfn 1767 . . . 4 𝑥 ¬ 𝜓
4 spcgf.3 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
54notbid 306 . . . 4 (𝑥 = 𝐴 → (¬ 𝜑 ↔ ¬ 𝜓))
61, 3, 5spcgf 3257 . . 3 (𝐴𝑉 → (∀𝑥 ¬ 𝜑 → ¬ 𝜓))
76con2d 127 . 2 (𝐴𝑉 → (𝜓 → ¬ ∀𝑥 ¬ 𝜑))
8 df-ex 1695 . 2 (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
97, 8syl6ibr 240 1 (𝐴𝑉 → (𝜓 → ∃𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wal 1472   = wceq 1474  wex 1694  wnf 1698  wcel 1976  wnfc 2734
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-v 3171
This theorem is referenced by:  spcegv  3263  rspce  3273  euotd  4888  bnj607  30043  bnj1491  30182  rspcegf  38005  rspcef  38067  stoweidlem36  38730  stoweidlem46  38740
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