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Theorem addneintr2d 7434
Description: Introducing a term on the right-hand side of a sum in a negated equality. Contrapositive of addcan2ad 7432. Consequence of addcan2d 7430. (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 )
addneintr2d.4  |-  ( ph  ->  A  =/=  B )
Assertion
Ref Expression
addneintr2d  |-  ( ph  ->  ( A  +  C
)  =/=  ( B  +  C ) )

Proof of Theorem addneintr2d
StepHypRef Expression
1 addneintr2d.4 . 2  |-  ( ph  ->  A  =/=  B )
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, 4addcan2d 7430 . . 3  |-  ( ph  ->  ( ( A  +  C )  =  ( B  +  C )  <-> 
A  =  B ) )
65necon3bid 2290 . 2  |-  ( ph  ->  ( ( A  +  C )  =/=  ( B  +  C )  <->  A  =/=  B ) )
71, 6mpbird 165 1  |-  ( ph  ->  ( A  +  C
)  =/=  ( B  +  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1434    =/= wne 2249  (class class class)co 5564   CCcc 7111    + caddc 7116
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-resscn 7200  ax-1cn 7201  ax-icn 7203  ax-addcl 7204  ax-addrcl 7205  ax-mulcl 7206  ax-addcom 7208  ax-addass 7210  ax-distr 7212  ax-i2m1 7213  ax-0id 7216  ax-rnegex 7217  ax-cnre 7219
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-iota 4917  df-fv 4960  df-ov 5567
This theorem is referenced by:  modsumfzodifsn  9548
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