Theorem add42i 8482

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 8481	. 2 |- ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( B + D ) )
6	2, 4	addcomi 8460	. . 3 |- ( B + D ) = ( D + B )
7	6	oveq2i 6086	. 2 |- ( ( A + C ) + ( B + D ) ) = ( ( A + C ) + ( D + B ) )
8	5, 7	eqtri 2259	1 |- ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( D + B ) )

Syntax hints:   = wceq 1402   e. wcel 2209  (class class class)co 6075   CCcc 8167   + caddc 8172

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-addcl 8265  ax-addcom 8269  ax-addass 8271

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-un 3224  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-iota 5332  df-fv 5380  df-ov 6078

This theorem is referenced by: (None)