Theorem cbvralsvwOLD 3292

Description: Obsolete version of cbvralsvw 3290 as of 21-Aug-2025. (Contributed by NM, 20-Nov-2005.) Avoid ax-13 2380. (Revised by GG, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 8-Mar-2025.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
cbvralsvwOLD	⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑)

Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem cbvralsvwOLD

Step	Hyp	Ref	Expression
1		nfv 1921	. 2 ⊢ Ⅎ𝑦𝜑
2		nfs1v 2167	. 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
3		sbequ12 2263	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	1, 2, 3	cbvralw 3281	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 207  [wsb 2073  ∀wral 3053

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-10 2152  ax-11 2168  ax-12 2189

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-ex 1787  df-nf 1791  df-sb 2074  df-clel 2814  df-nfc 2888  df-ral 3054

This theorem is referenced by: (None)