Theorem cbvrexsvw 3315

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 1947	. 2 ⊢ Ⅎ𝑦𝜑
2		nfs1v 2193	. 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
3		sbequ12 2287	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	1, 2, 3	cbvrexw 3306	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑)

This proof depends on syntax axioms:   ↔ wb 209  [wsb 2099  ∃wrex 3087

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-10 2178  ax-11 2194  ax-12 2213

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-sb 2100  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088

This theorem is used by:  rspesbca  3828  ac6sf  10560  ac6gf  38646