ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  poeq1 Unicode version

Theorem poeq1 4216
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 3926 . . . . . 6  |-  ( R  =  S  ->  (
x R x  <->  x S x ) )
21notbid 656 . . . . 5  |-  ( R  =  S  ->  ( -.  x R x  <->  -.  x S x ) )
3 breq 3926 . . . . . . 7  |-  ( R  =  S  ->  (
x R y  <->  x S
y ) )
4 breq 3926 . . . . . . 7  |-  ( R  =  S  ->  (
y R z  <->  y S
z ) )
53, 4anbi12d 464 . . . . . 6  |-  ( R  =  S  ->  (
( x R y  /\  y R z )  <->  ( x S y  /\  y S z ) ) )
6 breq 3926 . . . . . 6  |-  ( R  =  S  ->  (
x R z  <->  x S
z ) )
75, 6imbi12d 233 . . . . 5  |-  ( R  =  S  ->  (
( ( x R y  /\  y R z )  ->  x R z )  <->  ( (
x S y  /\  y S z )  ->  x S z ) ) )
82, 7anbi12d 464 . . . 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 2435 . . 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 2457 . 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 4213 . 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 4213 . 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 222 1  |-  ( R  =  S  ->  ( R  Po  A  <->  S  Po  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331   A.wral 2414   class class class wbr 3924    Po wpo 4211
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 603  ax-in2 604  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-cleq 2130  df-clel 2133  df-ral 2419  df-br 3925  df-po 4213
This theorem is referenced by:  soeq1  4232
  Copyright terms: Public domain W3C validator