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Theorem add4i 7895
Description: Rearrangement of 4 terms in a sum. (Contributed by NM, 9-May-1999.)
Hypotheses
Ref Expression
add.1 𝐴 ∈ ℂ
add.2 𝐵 ∈ ℂ
add.3 𝐶 ∈ ℂ
add4.4 𝐷 ∈ ℂ
Assertion
Ref Expression
add4i ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐵 + 𝐷))

Proof of Theorem add4i
StepHypRef Expression
1 add.1 . 2 𝐴 ∈ ℂ
2 add.2 . 2 𝐵 ∈ ℂ
3 add.3 . 2 𝐶 ∈ ℂ
4 add4.4 . 2 𝐷 ∈ ℂ
5 add4 7891 . 2 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐵 + 𝐷)))
61, 2, 3, 4, 5mp4an 423 1 ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐵 + 𝐷))
Colors of variables: wff set class
Syntax hints:   = wceq 1316  wcel 1465  (class class class)co 5742  cc 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-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-addcl 7684  ax-addcom 7688  ax-addass 7690
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-rex 2399  df-v 2662  df-un 3045  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:  add42i  7896  negdii  8014  numma  9193
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