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

Theorem cbvab 2157
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 1819 . . . 4 x[z / y]ψ
3 cbvab.1 . . . . . 6 yφ
4 cbvab.3 . . . . . . . 8 (x = y → (φψ))
54equcoms 1591 . . . . . . 7 (y = x → (φψ))
65bicomd 129 . . . . . 6 (y = x → (ψφ))
73, 6sbie 1671 . . . . 5 ([x / y]ψφ)
8 sbequ 1718 . . . . 5 (x = z → ([x / y]ψ ↔ [z / y]ψ))
97, 8syl5bbr 183 . . . 4 (x = z → (φ ↔ [z / y]ψ))
102, 9sbie 1671 . . 3 ([z / x]φ ↔ [z / y]ψ)
11 df-clab 2024 . . 3 (z {xφ} ↔ [z / x]φ)
12 df-clab 2024 . . 3 (z {yψ} ↔ [z / y]ψ)
1310, 11, 123bitr4i 201 . 2 (z {xφ} ↔ z {yψ})
1413eqriv 2034 1 {xφ} = {yψ}
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242  wnf 1346   wcel 1390  [wsb 1642  {cab 2023
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030
This theorem is referenced by:  cbvabv  2158  cbvrab  2549  cbvsbc  2785  cbvrabcsf  2905  dfdmf  4471  dfrnf  4518  funfvdm2f  5181  abrexex2g  5689  abrexex2  5693
  Copyright terms: Public domain W3C validator