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Theorem iuncom 4182
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 2887 . . . 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 4180 . . . . 5  |-  ( z  e.  U_ y  e.  B  C  <->  E. y  e.  B  z  e.  C )
32rexbii 2745 . . . 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 4180 . . . . 5  |-  ( z  e.  U_ x  e.  A  C  <->  E. x  e.  A  z  e.  C )
54rexbii 2745 . . . 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 277 . . 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 4180 . . 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 4180 . . 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 277 . 2  |-  ( z  e.  U_ x  e.  A  U_ y  e.  B  C  <->  z  e.  U_ y  e.  B  U_ x  e.  A  C
)
109eqriv 2440 1  |-  U_ x  e.  A  U_ y  e.  B  C  =  U_ y  e.  B  U_ x  e.  A  C
Colors of variables: wff setvar class
Syntax hints:    = wceq 1369    e. wcel 1756   E.wrex 2721   U_ciun 4176
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ral 2725  df-rex 2726  df-v 2979  df-iun 4178
This theorem is referenced by: (None)
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