Theorem cbvrexsvw 3316

Description: Change bound variable by using a substitution. Version of cbvrexsv 3364 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by NM, 2-Mar-2008.) Avoid ax-13 2372. (Revised by Gino Giotto, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 8-Mar-2025.)

Assertion

Ref	Expression
cbvrexsvw	⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑)

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

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem cbvrexsvw

Step	Hyp	Ref	Expression
1		nfv 1918	. 2 ⊢ Ⅎ𝑦𝜑
2		nfs1v 2154	. 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
3		sbequ12 2244	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	1, 2, 3	cbvrexw 3305	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 205  [wsb 2068  ∃wrex 3071

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-10 2138  ax-11 2155  ax-12 2172

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ex 1783  df-nf 1787  df-sb 2069  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072

This theorem is referenced by:  rspesbca  3876  ac6sf  10484  ac6gf  36600