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Theorem r19.3rzv 4172
 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 1956 . 2 𝑥𝜑
21r19.3rz 4170 1 (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥𝐴 𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ≠ wne 2896  ∀wral 3014  ∅c0 4023 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-v 3306  df-dif 3683  df-nul 4024 This theorem is referenced by:  r19.9rzv  4173  r19.37zv  4175  ralnralall  4188  iinconst  4638  cnvpo  5786  supicc  12434  coe1mul2lem1  19760  neipeltop  21056  utop3cls  22177  tgcgr4  25546  frgrregord013  27484  poimirlem23  33664  rencldnfi  37804  cvgdvgrat  38931
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