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Theorem add42d 8121
Description: Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro, 27-May-2016.)
Hypotheses
Ref Expression
addd.1 (𝜑𝐴 ∈ ℂ)
addd.2 (𝜑𝐵 ∈ ℂ)
addd.3 (𝜑𝐶 ∈ ℂ)
add4d.4 (𝜑𝐷 ∈ ℂ)
Assertion
Ref Expression
add42d (𝜑 → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐷 + 𝐵)))

Proof of Theorem add42d
StepHypRef Expression
1 addd.1 . 2 (𝜑𝐴 ∈ ℂ)
2 addd.2 . 2 (𝜑𝐵 ∈ ℂ)
3 addd.3 . 2 (𝜑𝐶 ∈ ℂ)
4 add4d.4 . 2 (𝜑𝐷 ∈ ℂ)
5 add42 8113 . 2 (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐷 + 𝐵)))
61, 2, 3, 4, 5syl22anc 1239 1 (𝜑 → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐷 + 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wcel 2148  (class class class)co 5870  cc 7804   + caddc 7809
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-addcl 7902  ax-addcom 7906  ax-addass 7908
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3809  df-br 4002  df-iota 5175  df-fv 5221  df-ov 5873
This theorem is referenced by:  remullem  10871  mul4sqlem  12381
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