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Theorem soeq1 4133
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  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 poeq1 4117 . . 3  |-  ( R  =  S  ->  ( R  Po  A  <->  S  Po  A ) )
2 breq 3839 . . . . . 6  |-  ( R  =  S  ->  (
x R y  <->  x S
y ) )
3 breq 3839 . . . . . . 7  |-  ( R  =  S  ->  (
x R z  <->  x S
z ) )
4 breq 3839 . . . . . . 7  |-  ( R  =  S  ->  (
z R y  <->  z S
y ) )
53, 4orbi12d 742 . . . . . 6  |-  ( R  =  S  ->  (
( x R z  \/  z R y )  <->  ( x S z  \/  z S y ) ) )
62, 5imbi12d 232 . . . . 5  |-  ( R  =  S  ->  (
( x R y  ->  ( x R z  \/  z R y ) )  <->  ( x S y  ->  (
x S z  \/  z S y ) ) ) )
762ralbidv 2402 . . . 4  |-  ( R  =  S  ->  ( A. y  e.  A  A. z  e.  A  ( x R y  ->  ( x R z  \/  z R y ) )  <->  A. y  e.  A  A. z  e.  A  ( x S y  ->  (
x S z  \/  z S y ) ) ) )
87ralbidv 2380 . . 3  |-  ( R  =  S  ->  ( A. x  e.  A  A. y  e.  A  A. z  e.  A  ( x R y  ->  ( x R z  \/  z R y ) )  <->  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( x S y  ->  (
x S z  \/  z S y ) ) ) )
91, 8anbi12d 457 . 2  |-  ( R  =  S  ->  (
( R  Po  A  /\  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( x R y  ->  ( x R z  \/  z R y ) ) )  <-> 
( S  Po  A  /\  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( x S y  ->  ( x S z  \/  z S y ) ) ) ) )
10 df-iso 4115 . 2  |-  ( R  Or  A  <->  ( R  Po  A  /\  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( x R y  ->  (
x R z  \/  z R y ) ) ) )
11 df-iso 4115 . 2  |-  ( S  Or  A  <->  ( S  Po  A  /\  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( x S y  ->  (
x S z  \/  z S y ) ) ) )
129, 10, 113bitr4g 221 1  |-  ( R  =  S  ->  ( R  Or  A  <->  S  Or  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664    = wceq 1289   A.wral 2359   class class class wbr 3837    Po wpo 4112    Or wor 4113
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-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-cleq 2081  df-clel 2084  df-ral 2364  df-br 3838  df-po 4114  df-iso 4115
This theorem is referenced by: (None)
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