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Theorem weeq1 4248
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 4236 . . 3  |-  ( R  =  S  ->  ( R  Fr  A  <->  S  Fr  A ) )
2 breq 3901 . . . . . . . 8  |-  ( R  =  S  ->  (
x R y  <->  x S
y ) )
3 breq 3901 . . . . . . . 8  |-  ( R  =  S  ->  (
y R z  <->  y S
z ) )
42, 3anbi12d 464 . . . . . . 7  |-  ( R  =  S  ->  (
( x R y  /\  y R z )  <->  ( x S y  /\  y S z ) ) )
5 breq 3901 . . . . . . 7  |-  ( R  =  S  ->  (
x R z  <->  x S
z ) )
64, 5imbi12d 233 . . . . . 6  |-  ( R  =  S  ->  (
( ( x R y  /\  y R z )  ->  x R z )  <->  ( (
x S y  /\  y S z )  ->  x S z ) ) )
76ralbidv 2414 . . . . 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 2414 . . . 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 2414 . . 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 464 . 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 4226 . 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 4226 . 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 222 1  |-  ( R  =  S  ->  ( R  We  A  <->  S  We  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1316   A.wral 2393   class class class wbr 3899    Fr wfr 4220    We wwe 4222
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-5 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-17 1491  ax-ial 1499  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-cleq 2110  df-clel 2113  df-ral 2398  df-br 3900  df-frfor 4223  df-frind 4224  df-wetr 4226
This theorem is referenced by: (None)
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