Theorem cbvrexsvw 3314

Description: Change bound variable by using a substitution. Version of cbvrexsv 3352 with a disjoint variable condition, which does not require ax-13 2401. (Contributed by NM, 2-Mar-2008.) Avoid ax-13 2401. (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 1946	. 2 ⊢ Ⅎ𝑦𝜑
2		nfs1v 2192	. 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
3		sbequ12 2286	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	1, 2, 3	cbvrexw 3305	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑)

This proof depends on syntax axioms:   ↔ wb 209  [wsb 2098  ∃wrex 3086

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-10 2177  ax-11 2193  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1812  df-nf 1816  df-sb 2099  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087

This theorem is used by:  rspesbca  3828  ac6sf  10572  ac6gf  38773