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

Theorem cbvab 2210
Description: Rule used to change bound variables, using implicit substitution. (Contributed by Andrew Salmon, 11-Jul-2011.)
Hypotheses
Ref Expression
cbvab.1 𝑦𝜑
cbvab.2 𝑥𝜓
cbvab.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvab {𝑥𝜑} = {𝑦𝜓}

Proof of Theorem cbvab
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 cbvab.2 . . . . 5 𝑥𝜓
21nfsb 1870 . . . 4 𝑥[𝑧 / 𝑦]𝜓
3 cbvab.1 . . . . . 6 𝑦𝜑
4 cbvab.3 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑𝜓))
54equcoms 1641 . . . . . . 7 (𝑦 = 𝑥 → (𝜑𝜓))
65bicomd 139 . . . . . 6 (𝑦 = 𝑥 → (𝜓𝜑))
73, 6sbie 1721 . . . . 5 ([𝑥 / 𝑦]𝜓𝜑)
8 sbequ 1768 . . . . 5 (𝑥 = 𝑧 → ([𝑥 / 𝑦]𝜓 ↔ [𝑧 / 𝑦]𝜓))
97, 8syl5bbr 192 . . . 4 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑦]𝜓))
102, 9sbie 1721 . . 3 ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
11 df-clab 2075 . . 3 (𝑧 ∈ {𝑥𝜑} ↔ [𝑧 / 𝑥]𝜑)
12 df-clab 2075 . . 3 (𝑧 ∈ {𝑦𝜓} ↔ [𝑧 / 𝑦]𝜓)
1310, 11, 123bitr4i 210 . 2 (𝑧 ∈ {𝑥𝜑} ↔ 𝑧 ∈ {𝑦𝜓})
1413eqriv 2085 1 {𝑥𝜑} = {𝑦𝜓}
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1289  wnf 1394  wcel 1438  [wsb 1692  {cab 2074
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081
This theorem is referenced by:  cbvabv  2211  cbvrab  2617  cbvsbc  2865  cbvrabcsf  2991  dfdmf  4617  dfrnf  4664  funfvdm2f  5353  abrexex2g  5873  abrexex2  5877
  Copyright terms: Public domain W3C validator