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Theorem spcimegf 2933
 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 (A V → (ψxφ))

Proof of Theorem spcimegf
StepHypRef Expression
1 spcimgf.1 . . . 4 xA
2 spcimgf.2 . . . . 5 xψ
32nfn 1793 . . . 4 x ¬ ψ
4 spcimegf.3 . . . . 5 (x = A → (ψφ))
54con3d 125 . . . 4 (x = A → (¬ φ → ¬ ψ))
61, 3, 5spcimgf 2932 . . 3 (A V → (x ¬ φ → ¬ ψ))
76con2d 107 . 2 (A V → (ψ → ¬ x ¬ φ))
8 df-ex 1542 . 2 (xφ ↔ ¬ x ¬ φ)
97, 8syl6ibr 218 1 (A V → (ψxφ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2476 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861 This theorem is referenced by: (None)
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