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Theorem addcanad 7666
Description: Cancelling a term on the left-hand side of a sum in an equality. Consequence of addcand 7664. (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 )
addcanad.4  |-  ( ph  ->  ( A  +  B
)  =  ( A  +  C ) )
Assertion
Ref Expression
addcanad  |-  ( ph  ->  B  =  C )

Proof of Theorem addcanad
StepHypRef Expression
1 addcanad.4 . 2  |-  ( ph  ->  ( A  +  B
)  =  ( A  +  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 )
52, 3, 4addcand 7664 . 2  |-  ( ph  ->  ( ( A  +  B )  =  ( A  +  C )  <-> 
B  =  C ) )
61, 5mpbid 145 1  |-  ( ph  ->  B  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    e. wcel 1438  (class class class)co 5652   CCcc 7346    + caddc 7351
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-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-resscn 7435  ax-1cn 7436  ax-icn 7438  ax-addcl 7439  ax-addrcl 7440  ax-mulcl 7441  ax-addcom 7443  ax-addass 7445  ax-distr 7447  ax-i2m1 7448  ax-0id 7451  ax-rnegex 7452  ax-cnre 7454
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-iota 4980  df-fv 5023  df-ov 5655
This theorem is referenced by:  divalglemqt  11193
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