Theorem addcan2d 8211

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 8207	. 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 2167  (class class class)co 5922   CCcc 7877   + caddc 7882

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-resscn 7971  ax-1cn 7972  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-iota 5219  df-fv 5266  df-ov 5925

This theorem is referenced by:  addcan2ad  8213  addneintr2d  8215  nn0opthd  10814