Theorem r19.3rzvOLD 4459

Description: Obsolete version of r19.3rzv 4458 as of 16-Feb-2026. (Contributed by NM, 10-Mar-1997.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
r19.3rzvOLD	⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥

Proof of Theorem r19.3rzvOLD

Step	Hyp	Ref	Expression
1		nfv 1916	. 2 ⊢ Ⅎ𝑥𝜑
2	1	r19.3rz 4456	1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))

Syntax hints:   → wi 4   ↔ wb 206   ≠ wne 2933  ∀wral 3052  ∅c0 4287

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-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-ne 2934  df-ral 3053  df-dif 3906  df-nul 4288

This theorem is referenced by: (None)