Theorem add42 7634

Description: Rearrangement of 4 terms in a sum. (Contributed by NM, 12-May-2005.)

Assertion

Ref	Expression
add42	|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( D + B ) ) )

Proof of Theorem add42

Step	Hyp	Ref	Expression
1		add4 7633	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( B + D ) ) )
2		addcom 7609	. . . 4 |- ( ( B e. CC /\ D e. CC ) -> ( B + D ) = ( D + B ) )
3	2	ad2ant2l 492	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( B + D ) = ( D + B ) )
4	3	oveq2d 5660	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + C ) + ( B + D ) ) = ( ( A + C ) + ( D + B ) ) )
5	1, 4	eqtrd 2120	1 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( D + B ) ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1289   e. wcel 1438  (class class class)co 5644   CCcc 7338   + caddc 7343

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-addcl 7431  ax-addcom 7435  ax-addass 7437

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-un 3003  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-br 3844  df-iota 4975  df-fv 5018  df-ov 5647

This theorem is referenced by:  add42d  7642