Theorem r19.3rzv 4497

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 1915	. 2 ⊢ Ⅎ𝑥𝜑
2	1	r19.3rz 4495	1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))

Syntax hints:   → wi 4   ↔ wb 205   ≠ wne 2938  ∀wral 3059  ∅c0 4321

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 1911  ax-6 1969  ax-7 2009  ax-9 2114  ax-12 2169  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-ne 2939  df-ral 3060  df-dif 3950  df-nul 4322

This theorem is referenced by:  r19.9rzv  4498  r19.37zv  4500  ralnralall  4517  iinconst  5006  cnvpo  6285  supicc  13482  coe1mul2lem1  22009  neipeltop  22853  utop3cls  23976  tgcgr4  28049  frgrregord013  29915  poimirlem23  36814  rencldnfi  41861  cvgdvgrat  43374