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Theorem r19.3rzv 4402
 Description: Restricted quantification of wff not containing quantified variable. (Contributed by NM, 10-Mar-1997.)
Assertion
Ref Expression
r19.3rzv (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥𝐴 𝜑))
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥

Proof of Theorem r19.3rzv
StepHypRef Expression
1 nfv 1915 . 2 𝑥𝜑
21r19.3rz 4400 1 (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥𝐴 𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ≠ wne 2987  ∀wral 3106  ∅c0 4243 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-ral 3111  df-dif 3884  df-nul 4244 This theorem is referenced by:  r19.9rzv  4403  r19.37zv  4405  ralnralall  4416  iinconst  4892  cnvpo  6107  supicc  12882  coe1mul2lem1  20906  neipeltop  21744  utop3cls  22867  tgcgr4  26335  frgrregord013  28190  poimirlem23  35099  rencldnfi  39805  cvgdvgrat  41060
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