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Theorem weeq2 4249
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 4238 . . 3  |-  ( A  =  B  ->  ( R  Fr  A  <->  R  Fr  B ) )
2 raleq 2603 . . . . 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 2611 . . . 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 2611 . . 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 464 . 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 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 ) ) )
7 df-wetr 4226 . 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 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-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-in 3047  df-ss 3054  df-frfor 4223  df-frind 4224  df-wetr 4226
This theorem is referenced by:  reg3exmid  4464
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