Theorem add42i 8064

Description: Rearrangement of 4 terms in a sum. (Contributed by NM, 22-Aug-1999.)

Hypotheses

Ref	Expression
add.1	|- A e. CC
add.2	|- B e. CC
add.3	|- C e. CC
add4.4	|- D e. CC

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 e. CC
2		add.2	. . 3 |- B e. CC
3		add.3	. . 3 |- C e. CC
4		add4.4	. . 3 |- D e. CC
5	1, 2, 3, 4	add4i 8063	. 2 |- ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( B + D ) )
6	2, 4	addcomi 8042	. . 3 |- ( B + D ) = ( D + B )
7	6	oveq2i 5853	. 2 |- ( ( A + C ) + ( B + D ) ) = ( ( A + C ) + ( D + B ) )
8	5, 7	eqtri 2186	1 |- ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( D + B ) )

Syntax hints:   = wceq 1343   e. wcel 2136  (class class class)co 5842   CCcc 7751   + caddc 7756

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-addcl 7849  ax-addcom 7853  ax-addass 7855

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-iota 5153  df-fv 5196  df-ov 5845

This theorem is referenced by: (None)