MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  soeq1 Unicode version

Theorem soeq1 4333
Description: Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.)
Assertion
Ref Expression
soeq1  |-  ( R  =  S  ->  ( R  Or  A  <->  S  Or  A ) )

Proof of Theorem soeq1
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 poeq1 4317 . . 3  |-  ( R  =  S  ->  ( R  Po  A  <->  S  Po  A ) )
2 breq 4025 . . . . 5  |-  ( R  =  S  ->  (
x R y  <->  x S
y ) )
3 biidd 228 . . . . 5  |-  ( R  =  S  ->  (
x  =  y  <->  x  =  y ) )
4 breq 4025 . . . . 5  |-  ( R  =  S  ->  (
y R x  <->  y S x ) )
52, 3, 43orbi123d 1251 . . . 4  |-  ( R  =  S  ->  (
( x R y  \/  x  =  y  \/  y R x )  <->  ( x S y  \/  x  =  y  \/  y S x ) ) )
652ralbidv 2585 . . 3  |-  ( R  =  S  ->  ( A. x  e.  A  A. y  e.  A  ( x R y  \/  x  =  y  \/  y R x )  <->  A. x  e.  A  A. y  e.  A  ( x S y  \/  x  =  y  \/  y S x ) ) )
71, 6anbi12d 691 . 2  |-  ( R  =  S  ->  (
( R  Po  A  /\  A. x  e.  A  A. y  e.  A  ( x R y  \/  x  =  y  \/  y R x ) )  <->  ( S  Po  A  /\  A. x  e.  A  A. y  e.  A  ( x S y  \/  x  =  y  \/  y S x ) ) ) )
8 df-so 4315 . 2  |-  ( R  Or  A  <->  ( R  Po  A  /\  A. x  e.  A  A. y  e.  A  ( x R y  \/  x  =  y  \/  y R x ) ) )
9 df-so 4315 . 2  |-  ( S  Or  A  <->  ( S  Po  A  /\  A. x  e.  A  A. y  e.  A  ( x S y  \/  x  =  y  \/  y S x ) ) )
107, 8, 93bitr4g 279 1  |-  ( R  =  S  ->  ( R  Or  A  <->  S  Or  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    \/ w3o 933    = wceq 1623   A.wral 2543   class class class wbr 4023    Po wpo 4312    Or wor 4313
This theorem is referenced by:  weeq1  4381  ltsopi  8512  cnso  12525  opsrtoslem2  16226  soeq12d  27134
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-ex 1529  df-nf 1532  df-cleq 2276  df-clel 2279  df-ral 2548  df-br 4024  df-po 4314  df-so 4315
  Copyright terms: Public domain W3C validator