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Theorem addneintr2d 7919
Description: Introducing a term on the right-hand side of a sum in a negated equality. Contrapositive of addcan2ad 7917. Consequence of addcan2d 7915. (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 7915 . . 3  |-  ( ph  ->  ( ( A  +  C )  =  ( B  +  C )  <-> 
A  =  B ) )
65necon3bid 2326 . 2  |-  ( ph  ->  ( ( A  +  C )  =/=  ( B  +  C )  <->  A  =/=  B ) )
71, 6mpbird 166 1  |-  ( ph  ->  ( A  +  C
)  =/=  ( B  +  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1465    =/= wne 2285  (class class class)co 5742   CCcc 7586    + caddc 7591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-resscn 7680  ax-1cn 7681  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-addass 7690  ax-distr 7692  ax-i2m1 7693  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-iota 5058  df-fv 5101  df-ov 5745
This theorem is referenced by:  modsumfzodifsn  10137
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