Theorem r19.3rzv 4459

Description: Restricted quantification of wff not containing quantified variable. (Contributed by NM, 10-Mar-1997.) Avoid ax-12 2214. (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 3155	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜑)
3		rspn0 4311	. 2 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑))
4	2, 3	impbid2 228	1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))

Syntax hints:   → wi 4   ↔ wb 208   ∈ wcel 2144   ≠ wne 2959  ∀wral 3078  ∅c0 4287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-ne 2960  df-ral 3079  df-dif 3909  df-nul 4288

This theorem is referenced by:  r19.9rzv  4461  r19.37zv  4463  ralnralall  4469  iinconst  4962  cnvpo  6276  supicc  13507  coe1mul2lem1  22332  neipeltop  23191  utop3cls  24313  tgcgr4  28702  frgrregord013  30599  poimirlem23  38147  rencldnfi  43403  cvgdvgrat  44894