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Theorem spcegf 3592
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 1855 . . . 4 𝑥 ¬ 𝜓
4 spcgf.3 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
54notbid 318 . . . 4 (𝑥 = 𝐴 → (¬ 𝜑 ↔ ¬ 𝜓))
61, 3, 5spcgf 3591 . . 3 (𝐴𝑉 → (∀𝑥 ¬ 𝜑 → ¬ 𝜓))
76con2d 134 . 2 (𝐴𝑉 → (𝜓 → ¬ ∀𝑥 ¬ 𝜑))
8 df-ex 1777 . 2 (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
97, 8imbitrrdi 252 1 (𝐴𝑉 → (𝜓 → ∃𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wal 1535   = wceq 1537  wex 1776  wnf 1780  wcel 2106  wnfc 2888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-cleq 2727  df-clel 2814  df-nfc 2890
This theorem is referenced by:  rspce  3611  euotd  5523  bnj607  34909  bnj1491  35050  rspcegf  44961  stoweidlem36  45992  stoweidlem46  46002  ichnreuop  47397  ichreuopeq  47398
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