Theorem r19.3rzv 4447

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 4445	1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))

Syntax hints:   → wi 4   ↔ wb 208   ≠ wne 3019  ∀wral 3141  ∅c0 4294

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-11 2160  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-dif 3942  df-nul 4295

This theorem is referenced by:  r19.9rzv  4448  r19.37zv  4450  ralnralall  4461  iinconst  4932  cnvpo  6141  supicc  12889  coe1mul2lem1  20438  neipeltop  21740  utop3cls  22863  tgcgr4  26320  frgrregord013  28177  poimirlem23  34919  rencldnfi  39424  cvgdvgrat  40651