Theorem cbvralsvw 3391

Description: Change bound variable by using a substitution. Version of cbvralsv 3393 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by NM, 20-Nov-2005.) (Revised by Gino Giotto, 10-Jan-2024.)

Assertion

Ref	Expression
cbvralsvw	⊢ (∀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 cbvralsvw

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1918	. . 3 ⊢ Ⅎ𝑧𝜑
2		nfs1v 2155	. . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
3		sbequ12 2247	. . 3 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
4	1, 2, 3	cbvralw 3363	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑧 ∈ 𝐴 [𝑧 / 𝑥]𝜑)
5		nfv 1918	. . 3 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑
6		nfv 1918	. . 3 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑
7		sbequ 2087	. . 3 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
8	5, 6, 7	cbvralw 3363	. 2 ⊢ (∀𝑧 ∈ 𝐴 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑)
9	4, 8	bitri 274	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 205  [wsb 2068  ∀wral 3063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-10 2139  ax-11 2156  ax-12 2173

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ex 1784  df-nf 1788  df-sb 2069  df-clel 2817  df-nfc 2888  df-ral 3068

This theorem is referenced by:  sbralie  3395  rspsbc  3808  ralxpf  5744  tfinds  7681  tfindes  7684  nn0min  31036