Theorem add32d 8239

Description: Commutative/associative law that swaps the last two terms in a triple 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 )

Assertion

Ref	Expression
add32d	|- ( ph -> ( ( A + B ) + C ) = ( ( A + C ) + B ) )

Proof of Theorem add32d

Step	Hyp	Ref	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		add32 8230	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( ( A + C ) + B ) )
5	1, 2, 3, 4	syl3anc 1249	1 |- ( ph -> ( ( A + B ) + C ) = ( ( A + C ) + B ) )

Syntax hints:   -> wi 4   = wceq 1372   e. wcel 2175  (class class class)co 5943   CCcc 7922   + caddc 7927

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186  ax-addcom 8024  ax-addass 8026

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-rex 2489  df-v 2773  df-un 3169  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-iota 5231  df-fv 5278  df-ov 5946

This theorem is referenced by:  nppcan  8293  muladd  8455  peano5uzti  9480  flqaddz  10438  seq3shft2  10624  zesq  10801  abstri  11357  bdtrilem  11492  pythagtriplem1  12530  pythagtriplem12  12540  gsumfzconst  13619