Theorem r19.3rzv 4429

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 1917	. 2 ⊢ Ⅎ𝑥𝜑
2	1	r19.3rz 4427	1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))

Syntax hints:   → wi 4   ↔ wb 205   ≠ wne 2943  ∀wral 3064  ∅c0 4256

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-ne 2944  df-ral 3069  df-dif 3890  df-nul 4257

This theorem is referenced by:  r19.9rzv  4430  r19.37zv  4432  ralnralall  4449  iinconst  4934  cnvpo  6190  supicc  13233  coe1mul2lem1  21438  neipeltop  22280  utop3cls  23403  tgcgr4  26892  frgrregord013  28759  poimirlem23  35800  rencldnfi  40643  cvgdvgrat  41931