ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  addcanad Unicode version

Theorem addcanad 7413
Description: Cancelling a term on the left-hand side of a sum in an equality. Consequence of addcand 7411. (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 7411 . 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 1285    e. wcel 1434  (class class class)co 5563   CCcc 7093    + caddc 7098
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 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 7182  ax-1cn 7183  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-addcom 7190  ax-addass 7192  ax-distr 7194  ax-i2m1 7195  ax-0id 7198  ax-rnegex 7199  ax-cnre 7201
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-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 5566
This theorem is referenced by:  divalglemqt  10526
  Copyright terms: Public domain W3C validator