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Theorem add42 7892
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
StepHypRef Expression
1 add4 7891 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  +  B )  +  ( C  +  D ) )  =  ( ( A  +  C )  +  ( B  +  D ) ) )
2 addcom 7867 . . . 4  |-  ( ( B  e.  CC  /\  D  e.  CC )  ->  ( B  +  D
)  =  ( D  +  B ) )
32ad2ant2l 499 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( B  +  D
)  =  ( D  +  B ) )
43oveq2d 5758 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  +  C )  +  ( B  +  D ) )  =  ( ( A  +  C )  +  ( D  +  B ) ) )
51, 4eqtrd 2150 1  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  +  B )  +  ( C  +  D ) )  =  ( ( A  +  C )  +  ( D  +  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316    e. wcel 1465  (class class class)co 5742   CCcc 7586    + caddc 7591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-addcl 7684  ax-addcom 7688  ax-addass 7690
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-iota 5058  df-fv 5101  df-ov 5745
This theorem is referenced by:  add42d  7900
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