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Theorem cbvrexsvw 3453
Description: Change bound variable by using a substitution. Version of cbvrexsv 3455 with a disjoint variable condition, which does not require ax-13 2392. (Contributed by NM, 2-Mar-2008.) (Revised by Gino Giotto, 10-Jan-2024.)
Assertion
Ref Expression
cbvrexsvw (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 [𝑦 / 𝑥]𝜑)
Distinct variable groups:   𝑥,𝐴   𝜑,𝑦   𝑦,𝐴,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem cbvrexsvw
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfv 1916 . . 3 𝑧𝜑
2 nfs1v 2161 . . 3 𝑥[𝑧 / 𝑥]𝜑
3 sbequ12 2255 . . 3 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
41, 2, 3cbvrexw 3426 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑧𝐴 [𝑧 / 𝑥]𝜑)
5 nfv 1916 . . 3 𝑦[𝑧 / 𝑥]𝜑
6 nfv 1916 . . 3 𝑧[𝑦 / 𝑥]𝜑
7 sbequ 2091 . . 3 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
85, 6, 7cbvrexw 3426 . 2 (∃𝑧𝐴 [𝑧 / 𝑥]𝜑 ↔ ∃𝑦𝐴 [𝑦 / 𝑥]𝜑)
94, 8bitri 278 1 (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 [𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 209  [wsb 2070  wrex 3133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-10 2146  ax-11 2162  ax-12 2179
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2071  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138
This theorem is referenced by:  rspesbca  3847  ac6sf  9896  ac6gf  35070
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