Theorem cbvralsvw 3308

Description: Change bound variable by using a substitution. Version of cbvralsv 3356 with a disjoint variable condition, which does not require ax-13 2365. (Contributed by NM, 20-Nov-2005.) Avoid ax-13 2365. (Revised by Gino Giotto, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 8-Mar-2025.)

Assertion

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

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

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem cbvralsvw

Step	Hyp	Ref	Expression
1		nfv 1909	. 2 ⊢ Ⅎ𝑦𝜑
2		nfs1v 2145	. 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
3		sbequ12 2235	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	1, 2, 3	cbvralw 3297	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 205  [wsb 2059  ∀wral 3055

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-10 2129  ax-11 2146  ax-12 2163

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ex 1774  df-nf 1778  df-sb 2060  df-clel 2804  df-nfc 2879  df-ral 3056

This theorem is referenced by:  sbralieALT  3349  rspsbc  3868  ralxpf  5840  tfinds  7846  tfindes  7849  nn0min  32531