Theorem addcan2i 8325

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

Step	Hyp	Ref	Expression
1		addcani.1	. 2 |- A e. CC
2		addcani.2	. 2 |- B e. CC
3		addcani.3	. 2 |- C e. CC
4		addcan2 8323	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + C ) = ( B + C ) <-> A = B ) )
5	1, 2, 3, 4	mp3an 1371	1 |- ( ( A + C ) = ( B + C ) <-> A = B )

Syntax hints:   <-> wb 105   = wceq 1395   e. wcel 2200  (class class class)co 6000   CCcc 7993   + caddc 7998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-resscn 8087  ax-1cn 8088  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-distr 8099  ax-i2m1 8100  ax-0id 8103  ax-rnegex 8104  ax-cnre 8106

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-iota 5277  df-fv 5325  df-ov 6003

This theorem is referenced by: (None)