Theorem r19.3rzv 4469

Description: Restricted quantification of wff not containing quantified variable. (Contributed by NM, 10-Mar-1997.) Avoid ax-12 2220. (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 3163	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜑)
3		rspn0 4319	. 2 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑))
4	2, 3	impbid2 229	1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))

Syntax hints:   → wi 4   ↔ wb 209   ∈ wcel 2150   ≠ wne 2965  ∀wral 3086  ∅c0 4294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-9 2160  ax-ext 2742

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-ne 2966  df-ral 3087  df-dif 3916  df-nul 4295

This theorem is referenced by:  r19.9rzv  4471  r19.37zv  4473  ralnralall  4479  iinconst  4972  cnvpo  6292  supicc  13531  coe1mul2lem1  22411  neipeltop  23269  utop3cls  24391  tgcgr4  28780  frgrregord013  30716  poimirlem23  38242  rencldnfi  43500  cvgdvgrat  44975