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Theorem cbvrexsvw 3318
Description: Change bound variable by using a substitution. Version of cbvrexsv 3367 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by NM, 2-Mar-2008.) Avoid ax-13 2377. (Revised by GG, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 8-Mar-2025.)
Assertion
Ref Expression
cbvrexsvw (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 [𝑦 / 𝑥]𝜑)
Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem cbvrexsvw
StepHypRef Expression
1 nfv 1914 . 2 𝑦𝜑
2 nfs1v 2156 . 2 𝑥[𝑦 / 𝑥]𝜑
3 sbequ12 2251 . 2 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
41, 2, 3cbvrexw 3307 1 (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 [𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 206  [wsb 2064  wrex 3070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-10 2141  ax-11 2157  ax-12 2177
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1780  df-nf 1784  df-sb 2065  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071
This theorem is referenced by:  rspesbca  3881  ac6sf  10529  ac6gf  37739
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