ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  poeq1 Unicode version

Theorem poeq1 4027
Description: Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.)
Assertion
Ref Expression
poeq1  R  S  R  Po  S  Po

Proof of Theorem poeq1
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq 3757 . . . . . 6  R  S  R  S
21notbid 591 . . . . 5  R  S  R  S
3 breq 3757 . . . . . . 7  R  S  R  S
4 breq 3757 . . . . . . 7  R  S  R  S
53, 4anbi12d 442 . . . . . 6  R  S  R  R  S  S
6 breq 3757 . . . . . 6  R  S  R  S
75, 6imbi12d 223 . . . . 5  R  S  R  R  R  S  S  S
82, 7anbi12d 442 . . . 4  R  S  R  R  R  R  S  S  S  S
98ralbidv 2320 . . 3  R  S  R  R  R  R  S  S  S  S
1092ralbidv 2342 . 2  R  S  R  R  R  R  S  S  S  S
11 df-po 4024 . 2  R  Po  R  R  R  R
12 df-po 4024 . 2  S  Po  S  S  S  S
1310, 11, 123bitr4g 212 1  R  S  R  Po  S  Po
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wa 97   wb 98   wceq 1242  wral 2300   class class class wbr 3755    Po wpo 4022
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-in1 544  ax-in2 545  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-cleq 2030  df-clel 2033  df-ral 2305  df-br 3756  df-po 4024
This theorem is referenced by:  soeq1  4043
  Copyright terms: Public domain W3C validator