MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  spcimegf Structured version   Visualization version   GIF version

Theorem spcimegf 3489
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 1902 . . . 4 𝑥 ¬ 𝜓
4 spcimegf.3 . . . . 5 (𝑥 = 𝐴 → (𝜓𝜑))
54con3d 150 . . . 4 (𝑥 = 𝐴 → (¬ 𝜑 → ¬ 𝜓))
61, 3, 5spcimgf 3488 . . 3 (𝐴𝑉 → (∀𝑥 ¬ 𝜑 → ¬ 𝜓))
76con2d 132 . 2 (𝐴𝑉 → (𝜓 → ¬ ∀𝑥 ¬ 𝜑))
8 df-ex 1824 . 2 (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
97, 8syl6ibr 244 1 (𝐴𝑉 → (𝜓 → ∃𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1599   = wceq 1601  wex 1823  wnf 1827  wcel 2107  wnfc 2919
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-v 3400
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator