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Theorem iuncom 4101
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 2871 . . . 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 4099 . . . . 5  |-  ( z  e.  U_ y  e.  B  C  <->  E. y  e.  B  z  e.  C )
32rexbii 2732 . . . 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 4099 . . . . 5  |-  ( z  e.  U_ x  e.  A  C  <->  E. x  e.  A  z  e.  C )
54rexbii 2732 . . . 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 270 . . 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 4099 . . 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 4099 . . 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 270 . 2  |-  ( z  e.  U_ x  e.  A  U_ y  e.  B  C  <->  z  e.  U_ y  e.  B  U_ x  e.  A  C
)
109eqriv 2435 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 1653    e. wcel 1726   E.wrex 2708   U_ciun 4095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-v 2960  df-iun 4097
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