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Theorem addcan2i 7257
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 7255 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  C
)  =  ( B  +  C )  <->  A  =  B ) )
51, 2, 3, 4mp3an 1243 1  |-  ( ( A  +  C )  =  ( B  +  C )  <->  A  =  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 102    = wceq 1259    e. wcel 1409  (class class class)co 5540   CCcc 6945    + caddc 6950
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-resscn 7034  ax-1cn 7035  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-addcom 7042  ax-addass 7044  ax-distr 7046  ax-i2m1 7047  ax-0id 7050  ax-rnegex 7051  ax-cnre 7053
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-iota 4895  df-fv 4938  df-ov 5543
This theorem is referenced by: (None)
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