ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  add32d Unicode version

Theorem add32d 7898
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
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 add32 7889 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  +  C )  =  ( ( A  +  C )  +  B ) )
51, 2, 3, 4syl3anc 1201 1  |-  ( ph  ->  ( ( A  +  B )  +  C
)  =  ( ( A  +  C )  +  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316    e. wcel 1465  (class class class)co 5742   CCcc 7586    + caddc 7591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-addcom 7688  ax-addass 7690
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-iota 5058  df-fv 5101  df-ov 5745
This theorem is referenced by:  nppcan  7952  muladd  8114  peano5uzti  9127  flqaddz  10038  seq3shft2  10214  zesq  10378  abstri  10844  bdtrilem  10978
  Copyright terms: Public domain W3C validator