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Theorem addcani 8157
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 8155 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶))
51, 2, 3, 4mp3an 1348 1 ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶)
Colors of variables: wff set class
Syntax hints:  wb 105   = wceq 1364  wcel 2160  (class class class)co 5891  cc 7827   + caddc 7832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-resscn 7921  ax-1cn 7922  ax-icn 7924  ax-addcl 7925  ax-addrcl 7926  ax-mulcl 7927  ax-addcom 7929  ax-addass 7931  ax-distr 7933  ax-i2m1 7934  ax-0id 7937  ax-rnegex 7938  ax-cnre 7940
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-iota 5193  df-fv 5239  df-ov 5894
This theorem is referenced by:  negdii  8259  fsumrelem  11497
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