Theorem r19.3rzv 4064

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 1843	. 2 ⊢ Ⅎ𝑥𝜑
2	1	r19.3rz 4062	1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))

Syntax hints:   → wi 4   ↔ wb 196   ≠ wne 2794  ∀wral 2912  ∅c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by:  r19.9rzv  4065  r19.37zv  4067  ralnralall  4080  iinconst  4530  cnvpo  5673  supicc  12320  coe1mul2lem1  19637  neipeltop  20933  utop3cls  22055  tgcgr4  25426  frgrregord013  27253  poimirlem23  33432  rencldnfi  37385  cvgdvgrat  38512