Theorem add42i 8345

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 8344	. 2 ⊢ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐵 + 𝐷))
6	2, 4	addcomi 8323	. . 3 ⊢ (𝐵 + 𝐷) = (𝐷 + 𝐵)
7	6	oveq2i 6029	. 2 ⊢ ((𝐴 + 𝐶) + (𝐵 + 𝐷)) = ((𝐴 + 𝐶) + (𝐷 + 𝐵))
8	5, 7	eqtri 2252	1 ⊢ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐷 + 𝐵))

Syntax hints:   = wceq 1397   ∈ wcel 2202  (class class class)co 6018  ℂcc 8030   + caddc 8035

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-addcl 8128  ax-addcom 8132  ax-addass 8134

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-iota 5286  df-fv 5334  df-ov 6021

This theorem is referenced by: (None)