Theorem r19.3rzv 4434

Description: Restricted quantification of wff not containing quantified variable. (Contributed by NM, 10-Mar-1997.) Avoid ax-12 2191. (Revised by TM, 16-Feb-2026.)

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		ax-1 6	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜑))
2	1	ralrimiv 3132	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜑)
3		rspn0 4287	. 2 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑))
4	2, 3	impbid2 228	1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))

Syntax hints:   → wi 4   ↔ wb 208   ∈ wcel 2121   ≠ wne 2936  ∀wral 3055  ∅c0 4264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-ne 2937  df-ral 3056  df-dif 3888  df-nul 4265

This theorem is referenced by:  r19.9rzv  4436  r19.37zv  4438  ralnralall  4444  iinconst  4935  cnvpo  6242  supicc  13449  coe1mul2lem1  22257  neipeltop  23116  utop3cls  24238  tgcgr4  28621  frgrregord013  30487  poimirlem23  38025  rencldnfi  43281  cvgdvgrat  44772