Theorem spcimegf 2934

Description: Existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
spcimgf.1	⊢ ℲxA
spcimgf.2	⊢ Ⅎxψ
spcimegf.3	⊢ (x = A → (ψ → φ))

Assertion

Ref	Expression
spcimegf	⊢ (A ∈ V → (ψ → ∃xφ))

Proof of Theorem spcimegf

Step	Hyp	Ref	Expression
1		spcimgf.1	. . . 4 ⊢ ℲxA
2		spcimgf.2	. . . . 5 ⊢ Ⅎxψ
3	2	nfn 1793	. . . 4 ⊢ Ⅎx ¬ ψ
4		spcimegf.3	. . . . 5 ⊢ (x = A → (ψ → φ))
5	4	con3d 125	. . . 4 ⊢ (x = A → (¬ φ → ¬ ψ))
6	1, 3, 5	spcimgf 2933	. . 3 ⊢ (A ∈ V → (∀x ¬ φ → ¬ ψ))
7	6	con2d 107	. 2 ⊢ (A ∈ V → (ψ → ¬ ∀x ¬ φ))
8		df-ex 1542	. 2 ⊢ (∃xφ ↔ ¬ ∀x ¬ φ)
9	7, 8	syl6ibr 218	1 ⊢ (A ∈ V → (ψ → ∃xφ))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2477

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 2479  df-v 2862

This theorem is referenced by: (None)