Theorem add42d 8168

Description: Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro, 27-May-2016.)

Hypotheses

Ref	Expression
addd.1	|- ( ph -> A e. CC )
addd.2	|- ( ph -> B e. CC )
addd.3	|- ( ph -> C e. CC )
add4d.4	|- ( ph -> D e. CC )

Assertion

Ref	Expression
add42d	|- ( ph -> ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( D + B ) ) )

Proof of Theorem add42d

Step	Hyp	Ref	Expression
1		addd.1	. 2 |- ( ph -> A e. CC )
2		addd.2	. 2 |- ( ph -> B e. CC )
3		addd.3	. 2 |- ( ph -> C e. CC )
4		add4d.4	. 2 |- ( ph -> D e. CC )
5		add42 8160	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( D + B ) ) )
6	1, 2, 3, 4, 5	syl22anc 1250	1 |- ( ph -> ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( D + B ) ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2160  (class class class)co 5904   CCcc 7849   + caddc 7854

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-addcl 7947  ax-addcom 7951  ax-addass 7953

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2758  df-un 3152  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3832  df-br 4026  df-iota 5203  df-fv 5250  df-ov 5907

This theorem is referenced by:  remullem  10927  mul4sqlem  12442