Theorem cbvralsvw 3314

Description: Change bound variable by using a substitution. Version of cbvralsv 3362 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by NM, 20-Nov-2005.) Avoid ax-13 2371. (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 1917	. 2 ⊢ Ⅎ𝑦𝜑
2		nfs1v 2153	. 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
3		sbequ12 2243	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	1, 2, 3	cbvralw 3303	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 205  [wsb 2067  ∀wral 3061

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-10 2137  ax-11 2154  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ex 1782  df-nf 1786  df-sb 2068  df-clel 2810  df-nfc 2885  df-ral 3062

This theorem is referenced by:  sbralieALT  3355  rspsbc  3873  ralxpf  5846  tfinds  7848  tfindes  7851  nn0min  32021