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Theorem r19.3rzv 4505
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 1912 . 2 𝑥𝜑
21r19.3rz 4503 1 (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥𝐴 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wne 2938  wral 3059  c0 4339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-9 2116  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-ne 2939  df-ral 3060  df-dif 3966  df-nul 4340
This theorem is referenced by:  r19.9rzv  4506  r19.37zv  4508  ralnralall  4521  iinconst  5007  cnvpo  6309  supicc  13538  coe1mul2lem1  22286  neipeltop  23153  utop3cls  24276  tgcgr4  28554  frgrregord013  30424  poimirlem23  37630  rencldnfi  42809  cvgdvgrat  44309
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