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 ↔ ∀𝑥 ∈ 𝐴 𝜑))

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥

Proof of Theorem r19.3rzv

Step	Hyp	Ref	Expression
1		nfv 1922	. 2 ⊢ Ⅎ𝑥𝜑
2	1	r19.3rz 4408	1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))

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