Theorem r19.3rzv 4465

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 1914	. 2 ⊢ Ⅎ𝑥𝜑
2	1	r19.3rz 4463	1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))

Syntax hints:   → wi 4   ↔ wb 206   ≠ wne 2926  ∀wral 3045  ∅c0 4299

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 2008  ax-9 2119  ax-12 2178  ax-ext 2702

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 2066  df-clab 2709  df-cleq 2722  df-ne 2927  df-ral 3046  df-dif 3920  df-nul 4300

This theorem is referenced by:  r19.9rzv  4466  r19.37zv  4468  ralnralall  4481  iinconst  4969  cnvpo  6263  supicc  13469  coe1mul2lem1  22160  neipeltop  23023  utop3cls  24146  tgcgr4  28465  frgrregord013  30331  poimirlem23  37644  rencldnfi  42816  cvgdvgrat  44309