Theorem addcan2ad 8230

Description: Cancelling a term on the right-hand side of a sum in an equality. Consequence of addcan2d 8228. (Contributed by David Moews, 28-Feb-2017.)

Hypotheses

Ref	Expression
addcand.1	|- ( ph -> A e. CC )
addcand.2	|- ( ph -> B e. CC )
addcand.3	|- ( ph -> C e. CC )
addcan2ad.4	|- ( ph -> ( A + C ) = ( B + C ) )

Assertion

Ref	Expression
addcan2ad	|- ( ph -> A = B )

Proof of Theorem addcan2ad

Step	Hyp	Ref	Expression
1		addcan2ad.4	. 2 |- ( ph -> ( A + C ) = ( B + C ) )
2		addcand.1	. . 3 |- ( ph -> A e. CC )
3		addcand.2	. . 3 |- ( ph -> B e. CC )
4		addcand.3	. . 3 |- ( ph -> C e. CC )
5	2, 3, 4	addcan2d 8228	. 2 |- ( ph -> ( ( A + C ) = ( B + C ) <-> A = B ) )
6	1, 5	mpbid 147	1 |- ( ph -> A = B )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167  (class class class)co 5925   CCcc 7894   + caddc 7899

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 7988  ax-1cn 7989  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-distr 8000  ax-i2m1 8001  ax-0id 8004  ax-rnegex 8005  ax-cnre 8007

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 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-iota 5220  df-fv 5267  df-ov 5928

This theorem is referenced by: (None)