Theorem cbvralsvwOLDOLD 3321

Description: Obsolete version of cbvralsvw 3318 as of 8-Mar-2025. (Contributed by NM, 20-Nov-2005.) Avoid ax-13 2374. (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 1913	. . 3 ⊢ Ⅎ𝑧𝜑
2		nfs1v 2152	. . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
3		sbequ12 2247	. . 3 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
4	1, 2, 3	cbvralw 3307	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑧 ∈ 𝐴 [𝑧 / 𝑥]𝜑)
5		nfv 1913	. . 3 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑
6		nfv 1913	. . 3 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑
7		sbequ 2083	. . 3 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
8	5, 6, 7	cbvralw 3307	. 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 2064  ∀wral 3063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-10 2136  ax-11 2153  ax-12 2173

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-ex 1778  df-nf 1782  df-sb 2065  df-clel 2813  df-nfc 2890  df-ral 3064

This theorem is referenced by: (None)