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

Theorem cbvsbcv 3758
 Description: Change the bound variable of a class substitution using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2382. Use the weaker cbvsbcvw 3756 when possible. (Contributed by NM, 30-Sep-2008.) (Revised by Mario Carneiro, 13-Oct-2016.) (New usage is discouraged.)
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 1915 . 2 𝑦𝜑
2 nfv 1915 . 2 𝑥𝜓
3 cbvsbcv.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
41, 2, 3cbvsbc 3757 1 ([𝐴 / 𝑥]𝜑[𝐴 / 𝑦]𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  [wsbc 3723 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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-13 2382  ax-ext 2773 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-clab 2780  df-cleq 2794  df-clel 2873  df-sbc 3724 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator