Theorem add4i 8272

Description: Rearrangement of 4 terms in a sum. (Contributed by NM, 9-May-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
add4i	|- ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( B + D ) )

Proof of Theorem add4i

Step	Hyp	Ref	Expression
1		add.1	. 2 |- A e. CC
2		add.2	. 2 |- B e. CC
3		add.3	. 2 |- C e. CC
4		add4.4	. 2 |- D e. CC
5		add4 8268	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( B + D ) ) )
6	1, 2, 3, 4, 5	mp4an 427	1 |- ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( B + D ) )

Syntax hints:   = wceq 1373   e. wcel 2178  (class class class)co 5967   CCcc 7958   + caddc 7963

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189  ax-addcl 8056  ax-addcom 8060  ax-addass 8062

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-iota 5251  df-fv 5298  df-ov 5970

This theorem is referenced by:  add42i  8273  negdii  8391  numma  9582