Theorem spcimegf 3260

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 1768	. . . 4 ⊢ Ⅎ𝑥 ¬ 𝜓
4		spcimegf.3	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))
5	4	con3d 147	. . . 4 ⊢ (𝑥 = 𝐴 → (¬ 𝜑 → ¬ 𝜓))
6	1, 3, 5	spcimgf 3259	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ¬ 𝜑 → ¬ 𝜓))
7	6	con2d 128	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ¬ ∀𝑥 ¬ 𝜑))
8		df-ex 1696	. 2 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
9	7, 8	syl6ibr 241	1 ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∃𝑥𝜑))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1473   = wceq 1475  ∃wex 1695  Ⅎwnf 1699   ∈ wcel 1977  Ⅎwnfc 2738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175

This theorem is referenced by:  bj-xnex  32245