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Theorem iuncom 3906
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 2653 . . . 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 3904 . . . . 5  |-  ( z  e.  U_ y  e.  B  C  <->  E. y  e.  B  z  e.  C )
32rexbii 2496 . . . 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 3904 . . . . 5  |-  ( z  e.  U_ x  e.  A  C  <->  E. x  e.  A  z  e.  C )
54rexbii 2496 . . . 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 212 . . 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 3904 . . 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 3904 . . 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 212 . 2  |-  ( z  e.  U_ x  e.  A  U_ y  e.  B  C  <->  z  e.  U_ y  e.  B  U_ x  e.  A  C
)
109eqriv 2185 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 1363    e. wcel 2159   E.wrex 2468   U_ciun 3900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ral 2472  df-rex 2473  df-v 2753  df-iun 3902
This theorem is referenced by: (None)
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