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Theorem soeq1 4205
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 4189 . . 3  |-  ( R  =  S  ->  ( R  Po  A  <->  S  Po  A ) )
2 breq 3899 . . . . . 6  |-  ( R  =  S  ->  (
x R y  <->  x S
y ) )
3 breq 3899 . . . . . . 7  |-  ( R  =  S  ->  (
x R z  <->  x S
z ) )
4 breq 3899 . . . . . . 7  |-  ( R  =  S  ->  (
z R y  <->  z S
y ) )
53, 4orbi12d 765 . . . . . 6  |-  ( R  =  S  ->  (
( x R z  \/  z R y )  <->  ( x S z  \/  z S y ) ) )
62, 5imbi12d 233 . . . . 5  |-  ( R  =  S  ->  (
( x R y  ->  ( x R z  \/  z R y ) )  <->  ( x S y  ->  (
x S z  \/  z S y ) ) ) )
762ralbidv 2434 . . . 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 2412 . . 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 462 . 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 4187 . 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 4187 . 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 222 1  |-  ( R  =  S  ->  ( R  Or  A  <->  S  Or  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 680    = wceq 1314   A.wral 2391   class class class wbr 3897    Po wpo 4184    Or wor 4185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-17 1489  ax-ial 1497  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-cleq 2108  df-clel 2111  df-ral 2396  df-br 3898  df-po 4186  df-iso 4187
This theorem is referenced by: (None)
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