MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbvrexsv Structured version   Visualization version   GIF version

Theorem cbvrexsv 3445
Description: Change bound variable by using a substitution. Usage of this theorem is discouraged because it depends on ax-13 2391. Use the weaker cbvrexsvw 3443 when possible. (Contributed by NM, 2-Mar-2008.) (Revised by Andrew Salmon, 11-Jul-2011.) (New usage is discouraged.)
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 1915 . . 3 𝑧𝜑
2 nfs1v 2160 . . 3 𝑥[𝑧 / 𝑥]𝜑
3 sbequ12 2254 . . 3 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
41, 2, 3cbvrex 3421 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑧𝐴 [𝑧 / 𝑥]𝜑)
5 nfv 1915 . . . 4 𝑦𝜑
65nfsb 2565 . . 3 𝑦[𝑧 / 𝑥]𝜑
7 nfv 1915 . . 3 𝑧[𝑦 / 𝑥]𝜑
8 sbequ 2090 . . 3 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
96, 7, 8cbvrex 3421 . 2 (∃𝑧𝐴 [𝑧 / 𝑥]𝜑 ↔ ∃𝑦𝐴 [𝑦 / 𝑥]𝜑)
104, 9bitri 278 1 (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 [𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 209  [wsb 2069  wrex 3131
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-10 2145  ax-11 2161  ax-12 2178  ax-13 2391
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136
This theorem is referenced by:  cbvexsv  41187
  Copyright terms: Public domain W3C validator