Theorem spcimegf 3489

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 1902	. . . 4 ⊢ Ⅎ𝑥 ¬ 𝜓
4		spcimegf.3	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))
5	4	con3d 150	. . . 4 ⊢ (𝑥 = 𝐴 → (¬ 𝜑 → ¬ 𝜓))
6	1, 3, 5	spcimgf 3488	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ¬ 𝜑 → ¬ 𝜓))
7	6	con2d 132	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ¬ ∀𝑥 ¬ 𝜑))
8		df-ex 1824	. 2 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
9	7, 8	syl6ibr 244	1 ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∃𝑥𝜑))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1599   = wceq 1601  ∃wex 1823  Ⅎwnf 1827   ∈ wcel 2107  Ⅎwnfc 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)