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Theorem r19.3rzv 4410
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 1922 . 2 𝑥𝜑
21r19.3rz 4408 1 (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥𝐴 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wne 2940  wral 3061  c0 4237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-9 2120  ax-12 2175  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-ne 2941  df-ral 3066  df-dif 3869  df-nul 4238
This theorem is referenced by:  r19.9rzv  4411  r19.37zv  4413  ralnralall  4430  iinconst  4914  cnvpo  6150  supicc  13089  coe1mul2lem1  21188  neipeltop  22026  utop3cls  23149  tgcgr4  26622  frgrregord013  28478  poimirlem23  35537  rencldnfi  40346  cvgdvgrat  41604
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