Theorem add42i 8097

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 8096	. 2 |- ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( B + D ) )
6	2, 4	addcomi 8075	. . 3 |- ( B + D ) = ( D + B )
7	6	oveq2i 5876	. 2 |- ( ( A + C ) + ( B + D ) ) = ( ( A + C ) + ( D + B ) )
8	5, 7	eqtri 2196	1 |- ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( D + B ) )

Syntax hints:   = wceq 1353   e. wcel 2146  (class class class)co 5865   CCcc 7784   + caddc 7789

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157  ax-addcl 7882  ax-addcom 7886  ax-addass 7888

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-rex 2459  df-v 2737  df-un 3131  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-iota 5170  df-fv 5216  df-ov 5868

This theorem is referenced by: (None)