Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  cbvoprab1 Unicode version

Theorem cbvoprab1 5607
 Description: Rule used to change first bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 5-Dec-2016.)
Hypotheses
Ref Expression
cbvoprab1.1
cbvoprab1.2
cbvoprab1.3
Assertion
Ref Expression
cbvoprab1
Distinct variable group:   ,,,
Allowed substitution hints:   (,,,)   (,,,)

Proof of Theorem cbvoprab1
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfv 1462 . . . . . 6
2 cbvoprab1.1 . . . . . 6
31, 2nfan 1498 . . . . 5
43nfex 1569 . . . 4
5 nfv 1462 . . . . . 6
6 cbvoprab1.2 . . . . . 6
75, 6nfan 1498 . . . . 5
87nfex 1569 . . . 4
9 opeq1 3578 . . . . . . 7
109eqeq2d 2093 . . . . . 6
11 cbvoprab1.3 . . . . . 6
1210, 11anbi12d 457 . . . . 5
1312exbidv 1747 . . . 4
144, 8, 13cbvex 1680 . . 3
1514opabbii 3853 . 2
16 dfoprab2 5583 . 2
17 dfoprab2 5583 . 2
1815, 16, 173eqtr4i 2112 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 102   wb 103   wceq 1285  wnf 1390  wex 1422  cop 3409  copab 3846  coprab 5544 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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-opab 3848  df-oprab 5547 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator