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Theorem iundif1 31891
Description: Indexed union of class difference with the subtrahend held constant. (Contributed by Brendan Leahy, 6-Aug-2018.)
Assertion
Ref Expression
iundif1  |-  U_ x  e.  A  ( B  \  C )  =  (
U_ x  e.  A  B  \  C )
Distinct variable group:    x, C
Allowed substitution hints:    A( x)    B( x)

Proof of Theorem iundif1
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 r19.41v 2977 . . . 4  |-  ( E. x  e.  A  ( y  e.  B  /\  -.  y  e.  C
)  <->  ( E. x  e.  A  y  e.  B  /\  -.  y  e.  C ) )
2 eldif 3446 . . . . 5  |-  ( y  e.  ( B  \  C )  <->  ( y  e.  B  /\  -.  y  e.  C ) )
32rexbii 2924 . . . 4  |-  ( E. x  e.  A  y  e.  ( B  \  C )  <->  E. x  e.  A  ( y  e.  B  /\  -.  y  e.  C ) )
4 eliun 4304 . . . . 5  |-  ( y  e.  U_ x  e.  A  B  <->  E. x  e.  A  y  e.  B )
54anbi1i 699 . . . 4  |-  ( ( y  e.  U_ x  e.  A  B  /\  -.  y  e.  C
)  <->  ( E. x  e.  A  y  e.  B  /\  -.  y  e.  C ) )
61, 3, 53bitr4i 280 . . 3  |-  ( E. x  e.  A  y  e.  ( B  \  C )  <->  ( y  e.  U_ x  e.  A  B  /\  -.  y  e.  C ) )
7 eliun 4304 . . 3  |-  ( y  e.  U_ x  e.  A  ( B  \  C )  <->  E. x  e.  A  y  e.  ( B  \  C ) )
8 eldif 3446 . . 3  |-  ( y  e.  ( U_ x  e.  A  B  \  C
)  <->  ( y  e. 
U_ x  e.  A  B  /\  -.  y  e.  C ) )
96, 7, 83bitr4i 280 . 2  |-  ( y  e.  U_ x  e.  A  ( B  \  C )  <->  y  e.  ( U_ x  e.  A  B  \  C ) )
109eqriv 2418 1  |-  U_ x  e.  A  ( B  \  C )  =  (
U_ x  e.  A  B  \  C )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 370    = wceq 1437    e. wcel 1872   E.wrex 2772    \ cdif 3433   U_ciun 4299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ral 2776  df-rex 2777  df-v 3082  df-dif 3439  df-iun 4301
This theorem is referenced by: (None)
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