Theorem cbvrexsvw 3313

Description: Change bound variable by using a substitution. Version of cbvrexsv 3353 with a disjoint variable condition, which does not require ax-13 2402. (Contributed by NM, 2-Mar-2008.) Avoid ax-13 2402. (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 1933	. 2 ⊢ Ⅎ𝑦𝜑
2		nfs1v 2189	. 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
3		sbequ12 2285	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	1, 2, 3	cbvrexw 3304	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 208  [wsb 2089  ∃wrex 3085

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-10 2174  ax-11 2190  ax-12 2211

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-ex 1799  df-nf 1803  df-sb 2090  df-clel 2836  df-nfc 2910  df-ral 3076  df-rex 3086

This theorem is referenced by:  rspesbca  3832  ac6sf  10440  ac6gf  38192