Theorem r19.3rzv 4455

Description: Restricted quantification of wff not containing quantified variable. (Contributed by NM, 10-Mar-1997.) Avoid ax-12 2183. (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 3126	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜑)
3		rspn0 4307	. 2 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑))
4	2, 3	impbid2 226	1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2114   ≠ wne 2931  ∀wral 3050  ∅c0 4284

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-ne 2932  df-ral 3051  df-dif 3903  df-nul 4285

This theorem is referenced by:  r19.9rzv  4457  r19.37zv  4459  ralnralall  4465  iinconst  4956  cnvpo  6244  supicc  13419  coe1mul2lem1  22211  neipeltop  23075  utop3cls  24197  tgcgr4  28584  frgrregord013  30451  poimirlem23  37813  rencldnfi  43100  cvgdvgrat  44591