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Theorem addcan2i 8058
Description: Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 14-May-2003.) (Revised by Scott Fenton, 3-Jan-2013.)
Hypotheses
Ref Expression
addcani.1  |-  A  e.  CC
addcani.2  |-  B  e.  CC
addcani.3  |-  C  e.  CC
Assertion
Ref Expression
addcan2i  |-  ( ( A  +  C )  =  ( B  +  C )  <->  A  =  B )

Proof of Theorem addcan2i
StepHypRef Expression
1 addcani.1 . 2  |-  A  e.  CC
2 addcani.2 . 2  |-  B  e.  CC
3 addcani.3 . 2  |-  C  e.  CC
4 addcan2 8056 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  C
)  =  ( B  +  C )  <->  A  =  B ) )
51, 2, 3, 4mp3an 1319 1  |-  ( ( A  +  C )  =  ( B  +  C )  <->  A  =  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    = wceq 1335    e. wcel 2128  (class class class)co 5824   CCcc 7730    + caddc 7735
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-resscn 7824  ax-1cn 7825  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-addcom 7832  ax-addass 7834  ax-distr 7836  ax-i2m1 7837  ax-0id 7840  ax-rnegex 7841  ax-cnre 7843
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-iota 5135  df-fv 5178  df-ov 5827
This theorem is referenced by: (None)
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