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Theorem add42d 7631
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
StepHypRef 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 7623 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  +  B )  +  ( C  +  D ) )  =  ( ( A  +  C )  +  ( D  +  B ) ) )
61, 2, 3, 4, 5syl22anc 1175 1  |-  ( ph  ->  ( ( A  +  B )  +  ( C  +  D ) )  =  ( ( A  +  C )  +  ( D  +  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    e. wcel 1438  (class class class)co 5634   CCcc 7327    + caddc 7332
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 7420  ax-addcom 7424  ax-addass 7426
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 3001  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-iota 4967  df-fv 5010  df-ov 5637
This theorem is referenced by:  remullem  10270
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