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Theorem poeq1 4317
Description: Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.)
Assertion
Ref Expression
poeq1  |-  ( R  =  S  ->  ( R  Po  A  <->  S  Po  A ) )

Proof of Theorem poeq1
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq 4025 . . . . . 6  |-  ( R  =  S  ->  (
x R x  <->  x S x ) )
21notbid 285 . . . . 5  |-  ( R  =  S  ->  ( -.  x R x  <->  -.  x S x ) )
3 breq 4025 . . . . . . 7  |-  ( R  =  S  ->  (
x R y  <->  x S
y ) )
4 breq 4025 . . . . . . 7  |-  ( R  =  S  ->  (
y R z  <->  y S
z ) )
53, 4anbi12d 691 . . . . . 6  |-  ( R  =  S  ->  (
( x R y  /\  y R z )  <->  ( x S y  /\  y S z ) ) )
6 breq 4025 . . . . . 6  |-  ( R  =  S  ->  (
x R z  <->  x S
z ) )
75, 6imbi12d 311 . . . . 5  |-  ( R  =  S  ->  (
( ( x R y  /\  y R z )  ->  x R z )  <->  ( (
x S y  /\  y S z )  ->  x S z ) ) )
82, 7anbi12d 691 . . . 4  |-  ( R  =  S  ->  (
( -.  x R x  /\  ( ( x R y  /\  y R z )  ->  x R z ) )  <-> 
( -.  x S x  /\  ( ( x S y  /\  y S z )  ->  x S z ) ) ) )
98ralbidv 2563 . . 3  |-  ( R  =  S  ->  ( A. z  e.  A  ( -.  x R x  /\  ( ( x R y  /\  y R z )  ->  x R z ) )  <->  A. z  e.  A  ( -.  x S x  /\  ( ( x S y  /\  y S z )  ->  x S z ) ) ) )
1092ralbidv 2585 . 2  |-  ( R  =  S  ->  ( A. x  e.  A  A. y  e.  A  A. z  e.  A  ( -.  x R x  /\  ( ( x R y  /\  y R z )  ->  x R z ) )  <->  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( -.  x S x  /\  ( ( x S y  /\  y S z )  ->  x S z ) ) ) )
11 df-po 4314 . 2  |-  ( R  Po  A  <->  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( -.  x R x  /\  (
( x R y  /\  y R z )  ->  x R
z ) ) )
12 df-po 4314 . 2  |-  ( S  Po  A  <->  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( -.  x S x  /\  (
( x S y  /\  y S z )  ->  x S
z ) ) )
1310, 11, 123bitr4g 279 1  |-  ( R  =  S  ->  ( R  Po  A  <->  S  Po  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623   A.wral 2543   class class class wbr 4023    Po wpo 4312
This theorem is referenced by:  soeq1  4333
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-an 360  df-ex 1529  df-nf 1532  df-cleq 2276  df-clel 2279  df-ral 2548  df-br 4024  df-po 4314
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