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Theorem spcimegf 3566
Description: Existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
spcimgf.1 𝑥𝐴
spcimgf.2 𝑥𝜓
spcimegf.3 (𝑥 = 𝐴 → (𝜓𝜑))
Assertion
Ref Expression
spcimegf (𝐴𝑉 → (𝜓 → ∃𝑥𝜑))

Proof of Theorem spcimegf
StepHypRef Expression
1 spcimgf.1 . . . 4 𝑥𝐴
2 spcimgf.2 . . . . 5 𝑥𝜓
32nfn 1858 . . . 4 𝑥 ¬ 𝜓
4 spcimegf.3 . . . . 5 (𝑥 = 𝐴 → (𝜓𝜑))
54con3d 155 . . . 4 (𝑥 = 𝐴 → (¬ 𝜑 → ¬ 𝜓))
61, 3, 5spcimgf 3565 . . 3 (𝐴𝑉 → (∀𝑥 ¬ 𝜑 → ¬ 𝜓))
76con2d 136 . 2 (𝐴𝑉 → (𝜓 → ¬ ∀𝑥 ¬ 𝜑))
8 df-ex 1782 . 2 (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
97, 8syl6ibr 255 1 (𝐴𝑉 → (𝜓 → ∃𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1536   = wceq 1538  wex 1781  wnf 1785  wcel 2115  wnfc 2958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-v 3473
This theorem is referenced by: (None)
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