Theorem addcan2ad 8158

Description: Cancelling a term on the right-hand side of a sum in an equality. Consequence of addcan2d 8156. (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 8156	. 2 |- ( ph -> ( ( A + C ) = ( B + C ) <-> A = B ) )
6	1, 5	mpbid 147	1 |- ( ph -> A = B )

Syntax hints:   -> wi 4   = wceq 1363   e. wcel 2158  (class class class)co 5888   CCcc 7823   + caddc 7828

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169  ax-resscn 7917  ax-1cn 7918  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-addcom 7925  ax-addass 7927  ax-distr 7929  ax-i2m1 7930  ax-0id 7933  ax-rnegex 7934  ax-cnre 7936

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-iota 5190  df-fv 5236  df-ov 5891

This theorem is referenced by: (None)