Theorem add42i 8125

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

Step	Hyp	Ref	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
5	1, 2, 3, 4	add4i 8124	. 2 ⊢ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐵 + 𝐷))
6	2, 4	addcomi 8103	. . 3 ⊢ (𝐵 + 𝐷) = (𝐷 + 𝐵)
7	6	oveq2i 5888	. 2 ⊢ ((𝐴 + 𝐶) + (𝐵 + 𝐷)) = ((𝐴 + 𝐶) + (𝐷 + 𝐵))
8	5, 7	eqtri 2198	1 ⊢ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐷 + 𝐵))

Syntax hints:   = wceq 1353   ∈ wcel 2148  (class class class)co 5877  ℂcc 7811   + caddc 7816

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 7909  ax-addcom 7913  ax-addass 7915

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 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-iota 5180  df-fv 5226  df-ov 5880

This theorem is referenced by: (None)