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Theorem cbvrexsv 2664
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 1508 . . 3 𝑧𝜑
2 nfs1v 1910 . . 3 𝑥[𝑧 / 𝑥]𝜑
3 sbequ12 1744 . . 3 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
41, 2, 3cbvrex 2649 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑧𝐴 [𝑧 / 𝑥]𝜑)
5 nfv 1508 . . . 4 𝑦𝜑
65nfsb 1917 . . 3 𝑦[𝑧 / 𝑥]𝜑
7 nfv 1508 . . 3 𝑧[𝑦 / 𝑥]𝜑
8 sbequ 1812 . . 3 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
96, 7, 8cbvrex 2649 . 2 (∃𝑧𝐴 [𝑧 / 𝑥]𝜑 ↔ ∃𝑦𝐴 [𝑦 / 𝑥]𝜑)
104, 9bitri 183 1 (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 [𝑦 / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  wb 104  [wsb 1735  wrex 2415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420
This theorem is referenced by:  rspesbca  2988  rexxpf  4681
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