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Theorem iuncom 3911
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 2701 . . . 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 3909 . . . . 5  |-  ( z  e.  U_ y  e.  B  C  <->  E. y  e.  B  z  e.  C )
32rexbii 2568 . . . 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 3909 . . . . 5  |-  ( z  e.  U_ x  e.  A  C  <->  E. x  e.  A  z  e.  C )
54rexbii 2568 . . . 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 3909 . . 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 3909 . . 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 2280 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 1623    e. wcel 1684   E.wrex 2544   U_ciun 3905
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-iun 3907
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