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Theorem add4d 8124
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
add4d  |-  ( ph  ->  ( ( A  +  B )  +  ( C  +  D ) )  =  ( ( A  +  C )  +  ( B  +  D ) ) )

Proof of Theorem add4d
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 add4 8116 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  -> 
( ( A  +  B )  +  ( C  +  D ) )  =  ( ( A  +  C )  +  ( B  +  D ) ) )
61, 2, 3, 4, 5syl22anc 1239 1  |-  ( ph  ->  ( ( A  +  B )  +  ( C  +  D ) )  =  ( ( A  +  C )  +  ( B  +  D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148  (class class class)co 5874   CCcc 7808    + caddc 7813
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-addcl 7906  ax-addcom 7910  ax-addass 7912
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-iota 5178  df-fv 5224  df-ov 5877
This theorem is referenced by:  apadd1  8563  binom3  10634  readd  10873  imadd  10881  max0addsup  11223  bdtri  11243  efi4p  11720  binom4  14290
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