Theorem r19.3rzvOLD 4458

Description: Obsolete version of r19.3rzv 4457 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 1934	. 2 ⊢ Ⅎ𝑥𝜑
2	1	r19.3rz 4455	1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))

Syntax hints:   → wi 4   ↔ wb 208   ≠ wne 2957  ∀wral 3076  ∅c0 4285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-9 2152  ax-12 2212  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-clab 2741  df-cleq 2754  df-ne 2958  df-ral 3077  df-dif 3907  df-nul 4286

This theorem is referenced by: (None)