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Theorem soeq1 4043
Description: Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.)
Assertion
Ref Expression
soeq1  R  S  R  Or  S  Or

Proof of Theorem soeq1
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 poeq1 4027 . . 3  R  S  R  Po  S  Po
2 breq 3757 . . . . . 6  R  S  R  S
3 breq 3757 . . . . . . 7  R  S  R  S
4 breq 3757 . . . . . . 7  R  S  R  S
53, 4orbi12d 706 . . . . . 6  R  S  R  R  S  S
62, 5imbi12d 223 . . . . 5  R  S  R  R  R  S  S  S
762ralbidv 2342 . . . 4  R  S  R  R  R  S  S  S
87ralbidv 2320 . . 3  R  S  R  R  R  S  S  S
91, 8anbi12d 442 . 2  R  S  R  Po  R  R  R  S  Po  S  S  S
10 df-iso 4025 . 2  R  Or  R  Po  R  R  R
11 df-iso 4025 . 2  S  Or  S  Po  S  S  S
129, 10, 113bitr4g 212 1  R  S  R  Or  S  Or
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wo 628   wceq 1242  wral 2300   class class class wbr 3755    Po wpo 4022    Or wor 4023
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-io 629  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  df-iso 4025
This theorem is referenced by: (None)
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