Theorem r19.3rzvOLD 4455

Description: Obsolete version of r19.3rzv 4454 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 1915	. 2 ⊢ Ⅎ𝑥𝜑
2	1	r19.3rz 4452	1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))

Syntax hints:   → wi 4   ↔ wb 206   ≠ wne 2930  ∀wral 3049  ∅c0 4283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-9 2123  ax-12 2182  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2713  df-cleq 2726  df-ne 2931  df-ral 3050  df-dif 3902  df-nul 4284

This theorem is referenced by: (None)