Theorem spcegf 3320

Description: Existential specialization, using implicit substitution. (Contributed by NM, 2-Feb-1997.)

Hypotheses

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

Assertion

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

Proof of Theorem spcegf

Step	Hyp	Ref	Expression
1		spcgf.1	. . . 4 ⊢ Ⅎ𝑥𝐴
2		spcgf.2	. . . . 5 ⊢ Ⅎ𝑥𝜓
3	2	nfn 1824	. . . 4 ⊢ Ⅎ𝑥 ¬ 𝜓
4		spcgf.3	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
5	4	notbid 307	. . . 4 ⊢ (𝑥 = 𝐴 → (¬ 𝜑 ↔ ¬ 𝜓))
6	1, 3, 5	spcgf 3319	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ¬ 𝜑 → ¬ 𝜓))
7	6	con2d 129	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ¬ ∀𝑥 ¬ 𝜑))
8		df-ex 1745	. 2 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
9	7, 8	syl6ibr 242	1 ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∃𝑥𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196  ∀wal 1521   = wceq 1523  ∃wex 1744  Ⅎwnf 1748   ∈ wcel 2030  Ⅎwnfc 2780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233

This theorem is referenced by:  spcegv  3325  rspce  3335  euotd  5004  bnj607  31112  bnj1491  31251  rspcegf  39496  stoweidlem36  40571  stoweidlem46  40581