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Theorem iuncom 3992
Description: Commutation of indexed unions. (Contributed by NM, 18-Dec-2008.)
Assertion
Ref Expression
iuncom  |-  U_ x  e.  A  U_ y  e.  B  C  =  U_ y  e.  B  U_ x  e.  A  C
Distinct variable groups:    y, A    x, B    x, y
Allowed substitution hints:    A( x)    B( y)    C( x, y)

Proof of Theorem iuncom
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 rexcom 2777 . . . 4  |-  ( E. x  e.  A  E. y  e.  B  z  e.  C  <->  E. y  e.  B  E. x  e.  A  z  e.  C )
2 eliun 3990 . . . . 5  |-  ( z  e.  U_ y  e.  B  C  <->  E. y  e.  B  z  e.  C )
32rexbii 2644 . . . 4  |-  ( E. x  e.  A  z  e.  U_ y  e.  B  C  <->  E. x  e.  A  E. y  e.  B  z  e.  C )
4 eliun 3990 . . . . 5  |-  ( z  e.  U_ x  e.  A  C  <->  E. x  e.  A  z  e.  C )
54rexbii 2644 . . . 4  |-  ( E. y  e.  B  z  e.  U_ x  e.  A  C  <->  E. y  e.  B  E. x  e.  A  z  e.  C )
61, 3, 53bitr4i 268 . . 3  |-  ( E. x  e.  A  z  e.  U_ y  e.  B  C  <->  E. y  e.  B  z  e.  U_ x  e.  A  C
)
7 eliun 3990 . . 3  |-  ( z  e.  U_ x  e.  A  U_ y  e.  B  C  <->  E. x  e.  A  z  e.  U_ y  e.  B  C
)
8 eliun 3990 . . 3  |-  ( z  e.  U_ y  e.  B  U_ x  e.  A  C  <->  E. y  e.  B  z  e.  U_ x  e.  A  C
)
96, 7, 83bitr4i 268 . 2  |-  ( z  e.  U_ x  e.  A  U_ y  e.  B  C  <->  z  e.  U_ y  e.  B  U_ x  e.  A  C
)
109eqriv 2355 1  |-  U_ x  e.  A  U_ y  e.  B  C  =  U_ y  e.  B  U_ x  e.  A  C
Colors of variables: wff set class
Syntax hints:    = wceq 1642    e. wcel 1710   E.wrex 2620   U_ciun 3986
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-rex 2625  df-v 2866  df-iun 3988
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