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Theorem risc 29992
Description: The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)
Assertion
Ref Expression
risc  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( R  ~=R  S  <->  E. f 
f  e.  ( R 
RngIso  S ) ) )
Distinct variable groups:    R, f    S, f

Proof of Theorem risc
StepHypRef Expression
1 isriscg 29990 . 2  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( R  ~=R  S  <->  ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  E. f  f  e.  ( R  RngIso  S ) ) ) )
21bianabs 878 1  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( R  ~=R  S  <->  E. f 
f  e.  ( R 
RngIso  S ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   E.wex 1596    e. wcel 1767   class class class wbr 4447  (class class class)co 6282   RingOpscrngo 25053    RngIso crngiso 29967    ~=R crisc 29968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-iota 5549  df-fv 5594  df-ov 6285  df-risc 29989
This theorem is referenced by:  risci  29993
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