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Theorem add42i 7952
Description: Rearrangement of 4 terms in a sum. (Contributed by NM, 22-Aug-1999.)
Hypotheses
Ref Expression
add.1 𝐴 ∈ ℂ
add.2 𝐵 ∈ ℂ
add.3 𝐶 ∈ ℂ
add4.4 𝐷 ∈ ℂ
Assertion
Ref Expression
add42i ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐷 + 𝐵))

Proof of Theorem add42i
StepHypRef Expression
1 add.1 . . 3 𝐴 ∈ ℂ
2 add.2 . . 3 𝐵 ∈ ℂ
3 add.3 . . 3 𝐶 ∈ ℂ
4 add4.4 . . 3 𝐷 ∈ ℂ
51, 2, 3, 4add4i 7951 . 2 ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐵 + 𝐷))
62, 4addcomi 7930 . . 3 (𝐵 + 𝐷) = (𝐷 + 𝐵)
76oveq2i 5793 . 2 ((𝐴 + 𝐶) + (𝐵 + 𝐷)) = ((𝐴 + 𝐶) + (𝐷 + 𝐵))
85, 7eqtri 2161 1 ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐷 + 𝐵))
Colors of variables: wff set class
Syntax hints:   = wceq 1332  wcel 1481  (class class class)co 5782  cc 7642   + caddc 7647
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-addcl 7740  ax-addcom 7744  ax-addass 7746
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-iota 5096  df-fv 5139  df-ov 5785
This theorem is referenced by: (None)
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