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Theorem addneintrd 7199
Description: Introducing a term on the left-hand side of a sum in a negated equality. Contrapositive of addcanad 7197. Consequence of addcand 7195. (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 )
addneintrd.4  |-  ( ph  ->  B  =/=  C )
Assertion
Ref Expression
addneintrd  |-  ( ph  ->  ( A  +  B
)  =/=  ( A  +  C ) )

Proof of Theorem addneintrd
StepHypRef Expression
1 addneintrd.4 . 2  |-  ( ph  ->  B  =/=  C )
2 addcand.1 . . . 4  |-  ( ph  ->  A  e.  CC )
3 addcand.2 . . . 4  |-  ( ph  ->  B  e.  CC )
4 addcand.3 . . . 4  |-  ( ph  ->  C  e.  CC )
52, 3, 4addcand 7195 . . 3  |-  ( ph  ->  ( ( A  +  B )  =  ( A  +  C )  <-> 
B  =  C ) )
65necon3bid 2246 . 2  |-  ( ph  ->  ( ( A  +  B )  =/=  ( A  +  C )  <->  B  =/=  C ) )
71, 6mpbird 156 1  |-  ( ph  ->  ( A  +  B
)  =/=  ( A  +  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1393    =/= wne 2204  (class class class)co 5512   CCcc 6887    + caddc 6892
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-resscn 6976  ax-1cn 6977  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-addass 6986  ax-distr 6988  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-cnre 6995
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910  df-ov 5515
This theorem is referenced by: (None)
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