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Theorem weeq2 4184
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 4173 . . 3  |-  ( A  =  B  ->  ( R  Fr  A  <->  R  Fr  B ) )
2 raleq 2562 . . . . 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 2570 . . . 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 2570 . . 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 457 . 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 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 ) ) )
7 df-wetr 4161 . 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 221 1  |-  ( A  =  B  ->  ( R  We  A  <->  R  We  B ) )
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-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-in 3005  df-ss 3012  df-frfor 4158  df-frind 4159  df-wetr 4161
This theorem is referenced by:  reg3exmid  4395
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