Theorem r19.3rzv 4451

Description: Restricted quantification of wff not containing quantified variable. (Contributed by NM, 10-Mar-1997.)

Assertion

Ref	Expression
r19.3rzv	⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥

Proof of Theorem r19.3rzv

Step	Hyp	Ref	Expression
1		nfv 1917	. 2 ⊢ Ⅎ𝑥𝜑
2	1	r19.3rz 4449	1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))

Syntax hints:   → wi 4   ↔ wb 205   ≠ wne 2941  ∀wral 3062  ∅c0 4277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-12 2171  ax-ext 2708

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2715  df-cleq 2729  df-ne 2942  df-ral 3063  df-dif 3908  df-nul 4278

This theorem is referenced by:  r19.9rzv  4452  r19.37zv  4454  ralnralall  4471  iinconst  4959  cnvpo  6232  supicc  13343  coe1mul2lem1  21548  neipeltop  22390  utop3cls  23513  tgcgr4  27247  frgrregord013  29113  poimirlem23  35956  rencldnfi  40956  cvgdvgrat  42304