Theorem add32d 8457

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 8448	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( ( A + C ) + B ) )
5	1, 2, 3, 4	syl3anc 1274	1 |- ( ph -> ( ( A + B ) + C ) = ( ( A + C ) + B ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205  (class class class)co 6058   CCcc 8141   + caddc 8146

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216  ax-addcom 8243  ax-addass 8245

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rex 2528  df-v 2817  df-un 3218  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-iota 5317  df-fv 5365  df-ov 6061

This theorem is referenced by:  nppcan  8511  muladd  8674  peano5uzti  9704  flqaddz  10681  seq3shft2  10867  zesq  11045  abstri  11814  bdtrilem  11949  pythagtriplem1  12988  pythagtriplem12  12998  gsumfzconst  14094