Theorem spcegf 3560

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 1884	. . . 4 ⊢ Ⅎ𝑥 ¬ 𝜓
4		spcgf.3	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
5	4	notbid 321	. . . 4 ⊢ (𝑥 = 𝐴 → (¬ 𝜑 ↔ ¬ 𝜓))
6	1, 3, 5	spcgf 3559	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ¬ 𝜑 → ¬ 𝜓))
7	6	con2d 135	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ¬ ∀𝑥 ¬ 𝜑))
8		df-ex 1807	. 2 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
9	7, 8	imbitrrdi 255	1 ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∃𝑥𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1565   = wceq 1567  ∃wex 1806  Ⅎwnf 1810   ∈ wcel 2149  Ⅎwnfc 2916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1807  df-nf 1811  df-cleq 2761  df-clel 2844  df-nfc 2918

This theorem is referenced by:  rspce  3579  euotd  5497  bnj607  35249  bnj1491  35390  permaxrep  45641  rspcegf  45669  stoweidlem36  46676  stoweidlem46  46686  ichnreuop  48144  ichreuopeq  48145