ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cbvsbcv GIF version

Theorem cbvsbcv 2910
Description: Change the bound variable of a class substitution using implicit substitution. (Contributed by NM, 30-Sep-2008.) (Revised by Mario Carneiro, 13-Oct-2016.)
Hypothesis
Ref Expression
cbvsbcv.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvsbcv ([𝐴 / 𝑥]𝜑[𝐴 / 𝑦]𝜓)
Distinct variable groups:   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem cbvsbcv
StepHypRef Expression
1 nfv 1493 . 2 𝑦𝜑
2 nfv 1493 . 2 𝑥𝜓
3 cbvsbcv.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
41, 2, 3cbvsbc 2909 1 ([𝐴 / 𝑥]𝜑[𝐴 / 𝑦]𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  [wsbc 2882
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-sbc 2883
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator