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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 ↔ ∀𝑥𝐴 𝜑))
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥

Proof of Theorem r19.3rzv
StepHypRef Expression
1 nfv 1914 . 2 𝑥𝜑
21r19.3rz 4497 1 (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥𝐴 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wne 2940  wral 3061  c0 4333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-ne 2941  df-ral 3062  df-dif 3954  df-nul 4334
This theorem is referenced by:  r19.9rzv  4500  r19.37zv  4502  ralnralall  4515  iinconst  5002  cnvpo  6307  supicc  13541  coe1mul2lem1  22270  neipeltop  23137  utop3cls  24260  tgcgr4  28539  frgrregord013  30414  poimirlem23  37650  rencldnfi  42832  cvgdvgrat  44332
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