Theorem addcand 8457

Description: Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Hypotheses

Ref	Expression
addcand.1	|- ( ph -> A e. CC )
addcand.2	|- ( ph -> B e. CC )
addcand.3	|- ( ph -> C e. CC )

Assertion

Ref	Expression
addcand	|- ( ph -> ( ( A + B ) = ( A + C ) <-> B = C ) )

Proof of Theorem addcand

Step	Hyp	Ref	Expression
1		addcand.1	. 2 |- ( ph -> A e. CC )
2		addcand.2	. 2 |- ( ph -> B e. CC )
3		addcand.3	. 2 |- ( ph -> C e. CC )
4		addcan 8453	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) = ( A + C ) <-> B = C ) )
5	1, 2, 3, 4	syl3anc 1274	1 |- ( ph -> ( ( A + B ) = ( A + C ) <-> B = C ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   e. wcel 2203  (class class class)co 6050   CCcc 8125   + caddc 8130

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214  ax-resscn 8219  ax-1cn 8220  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-iota 5312  df-fv 5360  df-ov 6053

This theorem is referenced by:  addcanad  8459  addneintrd  8461  negeu  8464  eqneg  9006  nn0opthd  11084  cjreb  11551