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Theorem weeq1 4183
Description: Equality theorem for the well-ordering predicate. (Contributed by NM, 9-Mar-1997.)
Assertion
Ref Expression
weeq1  |-  ( R  =  S  ->  ( R  We  A  <->  S  We  A ) )

Proof of Theorem weeq1
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 freq1 4171 . . 3  |-  ( R  =  S  ->  ( R  Fr  A  <->  S  Fr  A ) )
2 breq 3847 . . . . . . . 8  |-  ( R  =  S  ->  (
x R y  <->  x S
y ) )
3 breq 3847 . . . . . . . 8  |-  ( R  =  S  ->  (
y R z  <->  y S
z ) )
42, 3anbi12d 457 . . . . . . 7  |-  ( R  =  S  ->  (
( x R y  /\  y R z )  <->  ( x S y  /\  y S z ) ) )
5 breq 3847 . . . . . . 7  |-  ( R  =  S  ->  (
x R z  <->  x S
z ) )
64, 5imbi12d 232 . . . . . 6  |-  ( R  =  S  ->  (
( ( x R y  /\  y R z )  ->  x R z )  <->  ( (
x S y  /\  y S z )  ->  x S z ) ) )
76ralbidv 2380 . . . . 5  |-  ( R  =  S  ->  ( A. z  e.  A  ( ( x R y  /\  y R z )  ->  x R z )  <->  A. z  e.  A  ( (
x S y  /\  y S z )  ->  x S z ) ) )
87ralbidv 2380 . . . 4  |-  ( R  =  S  ->  ( A. y  e.  A  A. z  e.  A  ( ( x R y  /\  y R z )  ->  x R z )  <->  A. y  e.  A  A. z  e.  A  ( (
x S y  /\  y S z )  ->  x S z ) ) )
98ralbidv 2380 . . 3  |-  ( R  =  S  ->  ( A. x  e.  A  A. y  e.  A  A. z  e.  A  ( ( x R y  /\  y R z )  ->  x R z )  <->  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( (
x S y  /\  y S z )  ->  x S z ) ) )
101, 9anbi12d 457 . 2  |-  ( R  =  S  ->  (
( R  Fr  A  /\  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( ( x R y  /\  y R z )  ->  x R z ) )  <-> 
( S  Fr  A  /\  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( ( x S y  /\  y S z )  ->  x S z ) ) ) )
11 df-wetr 4161 . 2  |-  ( R  We  A  <->  ( R  Fr  A  /\  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( (
x R y  /\  y R z )  ->  x R z ) ) )
12 df-wetr 4161 . 2  |-  ( S  We  A  <->  ( S  Fr  A  /\  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( (
x S y  /\  y S z )  ->  x S z ) ) )
1310, 11, 123bitr4g 221 1  |-  ( R  =  S  ->  ( R  We  A  <->  S  We  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289   A.wral 2359   class class class wbr 3845    Fr wfr 4155    We wwe 4157
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-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 3846  df-frfor 4158  df-frind 4159  df-wetr 4161
This theorem is referenced by: (None)
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