Theorem spcegf 3552

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 1878	. . . 4 ⊢ Ⅎ𝑥 ¬ 𝜓
4		spcgf.3	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
5	4	notbid 320	. . . 4 ⊢ (𝑥 = 𝐴 → (¬ 𝜑 ↔ ¬ 𝜓))
6	1, 3, 5	spcgf 3551	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ¬ 𝜑 → ¬ 𝜓))
7	6	con2d 134	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ¬ ∀𝑥 ¬ 𝜑))
8		df-ex 1801	. 2 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
9	7, 8	imbitrrdi 254	1 ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∃𝑥𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208  ∀wal 1559   = wceq 1561  ∃wex 1800  Ⅎwnf 1804   ∈ wcel 2143  Ⅎwnfc 2910

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-ex 1801  df-nf 1805  df-cleq 2755  df-clel 2838  df-nfc 2912

This theorem is referenced by:  rspce  3571  euotd  5483  bnj607  35212  bnj1491  35353  permaxrep  45583  rspcegf  45604  stoweidlem36  46611  stoweidlem46  46621  ichnreuop  48079  ichreuopeq  48080