Theorem add42 8136

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 8135	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( B + D ) ) )
2		addcom 8111	. . . 4 |- ( ( B e. CC /\ D e. CC ) -> ( B + D ) = ( D + B ) )
3	2	ad2ant2l 508	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( B + D ) = ( D + B ) )
4	3	oveq2d 5906	. 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 2221	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 104   = wceq 1363   e. wcel 2159  (class class class)co 5890   CCcc 7826   + caddc 7831

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170  ax-addcl 7924  ax-addcom 7928  ax-addass 7930

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-rex 2473  df-v 2753  df-un 3147  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-br 4018  df-iota 5192  df-fv 5238  df-ov 5893

This theorem is referenced by:  add42d  8144