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Theorem limeq 4204
Description: Equality theorem for the limit predicate. (Contributed by NM, 22-Apr-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
limeq  |-  ( A  =  B  ->  ( Lim  A  <->  Lim  B ) )

Proof of Theorem limeq
StepHypRef Expression
1 ordeq 4199 . . 3  |-  ( A  =  B  ->  ( Ord  A  <->  Ord  B ) )
2 eleq2 2151 . . 3  |-  ( A  =  B  ->  ( (/) 
e.  A  <->  (/)  e.  B
) )
3 id 19 . . . 4  |-  ( A  =  B  ->  A  =  B )
4 unieq 3662 . . . 4  |-  ( A  =  B  ->  U. A  =  U. B )
53, 4eqeq12d 2102 . . 3  |-  ( A  =  B  ->  ( A  =  U. A  <->  B  =  U. B ) )
61, 2, 53anbi123d 1248 . 2  |-  ( A  =  B  ->  (
( Ord  A  /\  (/) 
e.  A  /\  A  =  U. A )  <->  ( Ord  B  /\  (/)  e.  B  /\  B  =  U. B ) ) )
7 dflim2 4197 . 2  |-  ( Lim 
A  <->  ( Ord  A  /\  (/)  e.  A  /\  A  =  U. A ) )
8 dflim2 4197 . 2  |-  ( Lim 
B  <->  ( Ord  B  /\  (/)  e.  B  /\  B  =  U. B ) )
96, 7, 83bitr4g 221 1  |-  ( A  =  B  ->  ( Lim  A  <->  Lim  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    /\ w3a 924    = wceq 1289    e. wcel 1438   (/)c0 3286   U.cuni 3653   Ord word 4189   Lim wlim 4191
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-3an 926  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-rex 2365  df-in 3005  df-ss 3012  df-uni 3654  df-tr 3937  df-iord 4193  df-ilim 4196
This theorem is referenced by:  limuni2  4224
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