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

Theorem cbvsbc 3593
Description: Change bound variables in a wff substitution. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Hypotheses
Ref Expression
cbvsbc.1 𝑦𝜑
cbvsbc.2 𝑥𝜓
cbvsbc.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvsbc ([𝐴 / 𝑥]𝜑[𝐴 / 𝑦]𝜓)

Proof of Theorem cbvsbc
StepHypRef Expression
1 cbvsbc.1 . . . 4 𝑦𝜑
2 cbvsbc.2 . . . 4 𝑥𝜓
3 cbvsbc.3 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
41, 2, 3cbvab 2872 . . 3 {𝑥𝜑} = {𝑦𝜓}
54eleq2i 2819 . 2 (𝐴 ∈ {𝑥𝜑} ↔ 𝐴 ∈ {𝑦𝜓})
6 df-sbc 3565 . 2 ([𝐴 / 𝑥]𝜑𝐴 ∈ {𝑥𝜑})
7 df-sbc 3565 . 2 ([𝐴 / 𝑦]𝜓𝐴 ∈ {𝑦𝜓})
85, 6, 73bitr4i 292 1 ([𝐴 / 𝑥]𝜑[𝐴 / 𝑦]𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wnf 1845  wcel 2127  {cab 2734  [wsbc 3564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-clab 2735  df-cleq 2741  df-clel 2744  df-sbc 3565
This theorem is referenced by:  cbvsbcv  3594  cbvcsb  3667
  Copyright terms: Public domain W3C validator