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Theorem cbvrexsvw 3316
Description: Change bound variable by using a substitution. Version of cbvrexsv 3365 with a disjoint variable condition, which does not require ax-13 2375. (Contributed by NM, 2-Mar-2008.) Avoid ax-13 2375. (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 1912 . 2 𝑦𝜑
2 nfs1v 2154 . 2 𝑥[𝑦 / 𝑥]𝜑
3 sbequ12 2249 . 2 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
41, 2, 3cbvrexw 3305 1 (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 [𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 206  [wsb 2062  wrex 3068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-10 2139  ax-11 2155  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-sb 2063  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069
This theorem is referenced by:  rspesbca  3890  ac6sf  10527  ac6gf  37719
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