Theorem r19.3rzvOLD 4463

Description: Obsolete version of r19.3rzv 4462 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 1948	. 2 ⊢ Ⅎ𝑥𝜑
2	1	r19.3rz 4460	1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ≠ wne 2959  ∀wral 3080  ∅c0 4281

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-9 2156  ax-12 2215  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-clab 2742  df-cleq 2755  df-ne 2960  df-ral 3081  df-dif 4050  df-nul 4282

This theorem is used by: (None)