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Theorem weeq2 4237
Description: Equality theorem for the well-ordering predicate. (Contributed by NM, 3-Apr-1994.)
Assertion
Ref Expression
weeq2  |-  ( A  =  B  ->  ( R  We  A  <->  R  We  B ) )

Proof of Theorem weeq2
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 freq2 4226 . . 3  |-  ( A  =  B  ->  ( R  Fr  A  <->  R  Fr  B ) )
2 raleq 2598 . . . . 5  |-  ( A  =  B  ->  ( A. z  e.  A  ( ( x R y  /\  y R z )  ->  x R z )  <->  A. z  e.  B  ( (
x R y  /\  y R z )  ->  x R z ) ) )
32raleqbi1dv 2606 . . . 4  |-  ( A  =  B  ->  ( A. y  e.  A  A. z  e.  A  ( ( x R y  /\  y R z )  ->  x R z )  <->  A. y  e.  B  A. z  e.  B  ( (
x R y  /\  y R z )  ->  x R z ) ) )
43raleqbi1dv 2606 . . 3  |-  ( A  =  B  ->  ( A. x  e.  A  A. y  e.  A  A. z  e.  A  ( ( x R y  /\  y R z )  ->  x R z )  <->  A. x  e.  B  A. y  e.  B  A. z  e.  B  ( (
x R y  /\  y R z )  ->  x R z ) ) )
51, 4anbi12d 462 . 2  |-  ( A  =  B  ->  (
( R  Fr  A  /\  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( ( x R y  /\  y R z )  ->  x R z ) )  <-> 
( R  Fr  B  /\  A. x  e.  B  A. y  e.  B  A. z  e.  B  ( ( x R y  /\  y R z )  ->  x R z ) ) ) )
6 df-wetr 4214 . 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 ) ) )
7 df-wetr 4214 . 2  |-  ( R  We  B  <->  ( R  Fr  B  /\  A. x  e.  B  A. y  e.  B  A. z  e.  B  ( (
x R y  /\  y R z )  ->  x R z ) ) )
85, 6, 73bitr4g 222 1  |-  ( A  =  B  ->  ( R  We  A  <->  R  We  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1312   A.wral 2388   class class class wbr 3893    Fr wfr 4208    We wwe 4210
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-in 3041  df-ss 3048  df-frfor 4211  df-frind 4212  df-wetr 4214
This theorem is referenced by:  reg3exmid  4452
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