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Theorem spcegf 3574
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 1852 . . . 4 𝑥 ¬ 𝜓
4 spcgf.3 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
54notbid 318 . . . 4 (𝑥 = 𝐴 → (¬ 𝜑 ↔ ¬ 𝜓))
61, 3, 5spcgf 3573 . . 3 (𝐴𝑉 → (∀𝑥 ¬ 𝜑 → ¬ 𝜓))
76con2d 134 . 2 (𝐴𝑉 → (𝜓 → ¬ ∀𝑥 ¬ 𝜑))
8 df-ex 1774 . 2 (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
97, 8imbitrrdi 251 1 (𝐴𝑉 → (𝜓 → ∃𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wal 1531   = wceq 1533  wex 1773  wnf 1777  wcel 2098  wnfc 2875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-v 3468
This theorem is referenced by:  rspce  3593  euotd  5504  bnj607  34446  bnj1491  34587  rspcegf  44257  stoweidlem36  45298  stoweidlem46  45308  ichnreuop  46686  ichreuopeq  46687
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