Theorem cbvralsvwOLDOLD 3292

Description: Obsolete version of cbvralsvw 3289 as of 8-Mar-2025. (Contributed by NM, 20-Nov-2005.) Avoid ax-13 2377. (Revised by GG, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

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

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem cbvralsvwOLDOLD

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1916	. . 3 ⊢ Ⅎ𝑧𝜑
2		nfs1v 2162	. . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
3		sbequ12 2259	. . 3 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
4	1, 2, 3	cbvralw 3280	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑧 ∈ 𝐴 [𝑧 / 𝑥]𝜑)
5		nfv 1916	. . 3 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑
6		nfv 1916	. . 3 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑
7		sbequ 2089	. . 3 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
8	5, 6, 7	cbvralw 3280	. 2 ⊢ (∀𝑧 ∈ 𝐴 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑)
9	4, 8	bitri 275	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 206  [wsb 2068  ∀wral 3052

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-8 2116  ax-10 2147  ax-11 2163  ax-12 2185

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1782  df-nf 1786  df-sb 2069  df-clel 2812  df-nfc 2886  df-ral 3053

This theorem is referenced by: (None)