Theorem r19.3rzv 4499

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 1918	. 2 ⊢ Ⅎ𝑥𝜑
2	1	r19.3rz 4497	1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))

Syntax hints:   → wi 4   ↔ wb 205   ≠ wne 2941  ∀wral 3062  ∅c0 4323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2117  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-ne 2942  df-ral 3063  df-dif 3952  df-nul 4324

This theorem is referenced by:  r19.9rzv  4500  r19.37zv  4502  ralnralall  4519  iinconst  5008  cnvpo  6287  supicc  13478  coe1mul2lem1  21789  neipeltop  22633  utop3cls  23756  tgcgr4  27782  frgrregord013  29648  poimirlem23  36511  rencldnfi  41559  cvgdvgrat  43072