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Theorem cbvrexsv 3181
Description: Change bound variable by using a substitution. (Contributed by NM, 2-Mar-2008.) (Revised by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
cbvrexsv (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 [𝑦 / 𝑥]𝜑)
Distinct variable groups:   𝑥,𝐴   𝜑,𝑦   𝑦,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem cbvrexsv
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfv 1842 . . 3 𝑧𝜑
2 nfs1v 2436 . . 3 𝑥[𝑧 / 𝑥]𝜑
3 sbequ12 2110 . . 3 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
41, 2, 3cbvrex 3166 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑧𝐴 [𝑧 / 𝑥]𝜑)
5 nfv 1842 . . . 4 𝑦𝜑
65nfsb 2439 . . 3 𝑦[𝑧 / 𝑥]𝜑
7 nfv 1842 . . 3 𝑧[𝑦 / 𝑥]𝜑
8 sbequ 2375 . . 3 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
96, 7, 8cbvrex 3166 . 2 (∃𝑧𝐴 [𝑧 / 𝑥]𝜑 ↔ ∃𝑦𝐴 [𝑦 / 𝑥]𝜑)
104, 9bitri 264 1 (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 [𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 196  [wsb 1879  wrex 2912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917
This theorem is referenced by:  rspesbca  3518  ac6sf  9308  ac6gf  33507  cbvexsv  38588
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