Theorem cbvrexsvw 3284

Description: Change bound variable by using a substitution. Version of cbvrexsv 3333 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by NM, 2-Mar-2008.) Avoid ax-13 2372. (Revised by GG, 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 1915	. 2 ⊢ Ⅎ𝑦𝜑
2		nfs1v 2159	. 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
3		sbequ12 2254	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	1, 2, 3	cbvrexw 3275	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 206  [wsb 2067  ∃wrex 3056

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-10 2144  ax-11 2160  ax-12 2180

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1781  df-nf 1785  df-sb 2068  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057

This theorem is referenced by:  rspesbca  3827  ac6sf  10375  ac6gf  37772