Theorem r19.3rzv 4503

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 1910	. 2 ⊢ Ⅎ𝑥𝜑
2	1	r19.3rz 4501	1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))

Syntax hints:   → wi 4   ↔ wb 205   ≠ wne 2930  ∀wral 3051  ∅c0 4325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-9 2109  ax-12 2167  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-ne 2931  df-ral 3052  df-dif 3950  df-nul 4326

This theorem is referenced by:  r19.9rzv  4504  r19.37zv  4506  ralnralall  4523  iinconst  5013  cnvpo  6300  supicc  13534  coe1mul2lem1  22260  neipeltop  23127  utop3cls  24250  tgcgr4  28461  frgrregord013  30331  poimirlem23  37346  rencldnfi  42496  cvgdvgrat  44005