Theorem r19.3rzv 4463

Description: Restricted quantification of wff not containing quantified variable. (Contributed by NM, 10-Mar-1997.) Avoid ax-12 2212. (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 4310	. 2 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑))
4	2, 3	impbid2 229	1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∈ wcel 2142   ≠ wne 2957  ∀wral 3078  ∅c0 4285

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-ne 2958  df-ral 3079  df-dif 3907  df-nul 4286

This theorem is used by:  r19.9rzv  4465  r19.37zv  4467  ralnralall  4473  iinconst  4966  cnvpo  6288  supicc  13534  coe1mul2lem1  22439  neipeltop  23297  utop3cls  24419  tgcgr4  28811  frgrregord013  30757  poimirlem23  38322  rencldnfi  43576  cvgdvgrat  45051