MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  equncom Unicode version

Theorem equncom 3333
Description: If a class equals the union of two other classes, then it equals the union of those two classes commuted. equncom 3333 was automatically derived from equncomVD 28960 using the tools program translatewithout_overwriting.cmd and minimizing. (Contributed by Alan Sare, 18-Feb-2012.)
Assertion
Ref Expression
equncom  |-  ( A  =  ( B  u.  C )  <->  A  =  ( C  u.  B
) )

Proof of Theorem equncom
StepHypRef Expression
1 uncom 3332 . 2  |-  ( B  u.  C )  =  ( C  u.  B
)
21eqeq2i 2306 1  |-  ( A  =  ( B  u.  C )  <->  A  =  ( C  u.  B
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1632    u. cun 3163
This theorem is referenced by:  equncomi  3334  splint  25151  equncomiVD  28961
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170
  Copyright terms: Public domain W3C validator