Theorem r19.3rzv 4358

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 1892	. 2 ⊢ Ⅎ𝑥𝜑
2	1	r19.3rz 4356	1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))

Syntax hints:   → wi 4   ↔ wb 207   ≠ wne 2984  ∀wral 3105  ∅c0 4211

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-11 2126  ax-12 2141  ax-ext 2769

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-dif 3862  df-nul 4212

This theorem is referenced by:  r19.9rzv  4359  r19.37zv  4361  ralnralall  4372  iinconst  4835  cnvpo  6013  supicc  12736  coe1mul2lem1  20118  neipeltop  21421  utop3cls  22543  tgcgr4  25999  frgrregord013  27866  poimirlem23  34446  rencldnfi  38903  cvgdvgrat  40183