Theorem r19.3rzv 4522

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 1913	. 2 ⊢ Ⅎ𝑥𝜑
2	1	r19.3rz 4520	1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))

Syntax hints:   → wi 4   ↔ wb 206   ≠ wne 2946  ∀wral 3067  ∅c0 4352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-9 2118  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-ne 2947  df-ral 3068  df-dif 3979  df-nul 4353

This theorem is referenced by:  r19.9rzv  4523  r19.37zv  4525  ralnralall  4538  iinconst  5025  cnvpo  6318  supicc  13561  coe1mul2lem1  22291  neipeltop  23158  utop3cls  24281  tgcgr4  28557  frgrregord013  30427  poimirlem23  37603  rencldnfi  42777  cvgdvgrat  44282