Theorem addcan2d 8194

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 8190	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + C ) = ( B + C ) <-> A = B ) )
5	1, 2, 3, 4	syl3anc 1249	1 |- ( ph -> ( ( A + C ) = ( B + C ) <-> A = B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   e. wcel 2164  (class class class)co 5910   CCcc 7860   + caddc 7865

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-resscn 7954  ax-1cn 7955  ax-icn 7957  ax-addcl 7958  ax-addrcl 7959  ax-mulcl 7960  ax-addcom 7962  ax-addass 7964  ax-distr 7966  ax-i2m1 7967  ax-0id 7970  ax-rnegex 7971  ax-cnre 7973

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-iota 5207  df-fv 5254  df-ov 5913

This theorem is referenced by:  addcan2ad  8196  addneintr2d  8198  nn0opthd  10783