Theorem addcan2d 8319

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
addcan2d	|- ( ph -> ( ( A + C ) = ( B + C ) <-> A = B ) )

Proof of Theorem addcan2d

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		addcan2 8315	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + C ) = ( B + C ) <-> A = B ) )
5	1, 2, 3, 4	syl3anc 1271	1 |- ( ph -> ( ( A + C ) = ( B + C ) <-> A = B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   e. wcel 2200  (class class class)co 5994   CCcc 7985   + caddc 7990

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 8079  ax-1cn 8080  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-addcom 8087  ax-addass 8089  ax-distr 8091  ax-i2m1 8092  ax-0id 8095  ax-rnegex 8096  ax-cnre 8098

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 5274  df-fv 5322  df-ov 5997

This theorem is referenced by:  addcan2ad  8321  addneintr2d  8323  nn0opthd  10931  ccatws1lenp1bg  11154  ccatopth2  11235