Theorem add32d 8260

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 8251	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( ( A + C ) + B ) )
5	1, 2, 3, 4	syl3anc 1250	1 |- ( ph -> ( ( A + B ) + C ) = ( ( A + C ) + B ) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2177  (class class class)co 5957   CCcc 7943   + caddc 7948

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188  ax-addcom 8045  ax-addass 8047

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-rex 2491  df-v 2775  df-un 3174  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-br 4052  df-iota 5241  df-fv 5288  df-ov 5960

This theorem is referenced by:  nppcan  8314  muladd  8476  peano5uzti  9501  flqaddz  10462  seq3shft2  10648  zesq  10825  abstri  11490  bdtrilem  11625  pythagtriplem1  12663  pythagtriplem12  12673  gsumfzconst  13752