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Theorem addcani 7718
Description: Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 27-Oct-1999.) (Revised by Scott Fenton, 3-Jan-2013.)
Hypotheses
Ref Expression
addcani.1 𝐴 ∈ ℂ
addcani.2 𝐵 ∈ ℂ
addcani.3 𝐶 ∈ ℂ
Assertion
Ref Expression
addcani ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶)

Proof of Theorem addcani
StepHypRef Expression
1 addcani.1 . 2 𝐴 ∈ ℂ
2 addcani.2 . 2 𝐵 ∈ ℂ
3 addcani.3 . 2 𝐶 ∈ ℂ
4 addcan 7716 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶))
51, 2, 3, 4mp3an 1274 1 ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶)
Colors of variables: wff set class
Syntax hints:  wb 104   = wceq 1290  wcel 1439  (class class class)co 5666  cc 7402   + caddc 7407
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-resscn 7491  ax-1cn 7492  ax-icn 7494  ax-addcl 7495  ax-addrcl 7496  ax-mulcl 7497  ax-addcom 7499  ax-addass 7501  ax-distr 7503  ax-i2m1 7504  ax-0id 7507  ax-rnegex 7508  ax-cnre 7510
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-iota 4993  df-fv 5036  df-ov 5669
This theorem is referenced by:  negdii  7820  fsumrelem  10919
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