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Theorem poeq1 4644
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 4294 . . . . . 6  |-  ( R  =  S  ->  (
x R x  <->  x S x ) )
21notbid 294 . . . . 5  |-  ( R  =  S  ->  ( -.  x R x  <->  -.  x S x ) )
3 breq 4294 . . . . . . 7  |-  ( R  =  S  ->  (
x R y  <->  x S
y ) )
4 breq 4294 . . . . . . 7  |-  ( R  =  S  ->  (
y R z  <->  y S
z ) )
53, 4anbi12d 710 . . . . . 6  |-  ( R  =  S  ->  (
( x R y  /\  y R z )  <->  ( x S y  /\  y S z ) ) )
6 breq 4294 . . . . . 6  |-  ( R  =  S  ->  (
x R z  <->  x S
z ) )
75, 6imbi12d 320 . . . . 5  |-  ( R  =  S  ->  (
( ( x R y  /\  y R z )  ->  x R z )  <->  ( (
x S y  /\  y S z )  ->  x S z ) ) )
82, 7anbi12d 710 . . . 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 2735 . . 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 2757 . 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 4641 . 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 4641 . 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 288 1  |-  ( R  =  S  ->  ( R  Po  A  <->  S  Po  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369   A.wral 2715   class class class wbr 4292    Po wpo 4639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-12 1792  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1587  df-nf 1590  df-cleq 2436  df-clel 2439  df-ral 2720  df-br 4293  df-po 4641
This theorem is referenced by:  soeq1  4660
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