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Theorem iundif2 4398
Description: Indexed union of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use intiin 4385 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.)
Assertion
Ref Expression
iundif2  |-  U_ x  e.  A  ( B  \  C )  =  ( B  \  |^|_ x  e.  A  C )
Distinct variable group:    x, B
Allowed substitution hints:    A( x)    C( x)

Proof of Theorem iundif2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eldif 3491 . . . . 5  |-  ( y  e.  ( B  \  C )  <->  ( y  e.  B  /\  -.  y  e.  C ) )
21rexbii 2969 . . . 4  |-  ( E. x  e.  A  y  e.  ( B  \  C )  <->  E. x  e.  A  ( y  e.  B  /\  -.  y  e.  C ) )
3 r19.42v 3021 . . . 4  |-  ( E. x  e.  A  ( y  e.  B  /\  -.  y  e.  C
)  <->  ( y  e.  B  /\  E. x  e.  A  -.  y  e.  C ) )
4 rexnal 2915 . . . . . 6  |-  ( E. x  e.  A  -.  y  e.  C  <->  -.  A. x  e.  A  y  e.  C )
5 vex 3121 . . . . . . 7  |-  y  e. 
_V
6 eliin 4337 . . . . . . 7  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  A  C  <->  A. x  e.  A  y  e.  C ) )
75, 6ax-mp 5 . . . . . 6  |-  ( y  e.  |^|_ x  e.  A  C 
<-> 
A. x  e.  A  y  e.  C )
84, 7xchbinxr 311 . . . . 5  |-  ( E. x  e.  A  -.  y  e.  C  <->  -.  y  e.  |^|_ x  e.  A  C )
98anbi2i 694 . . . 4  |-  ( ( y  e.  B  /\  E. x  e.  A  -.  y  e.  C )  <->  ( y  e.  B  /\  -.  y  e.  |^|_ x  e.  A  C )
)
102, 3, 93bitri 271 . . 3  |-  ( E. x  e.  A  y  e.  ( B  \  C )  <->  ( y  e.  B  /\  -.  y  e.  |^|_ x  e.  A  C ) )
11 eliun 4336 . . 3  |-  ( y  e.  U_ x  e.  A  ( B  \  C )  <->  E. x  e.  A  y  e.  ( B  \  C ) )
12 eldif 3491 . . 3  |-  ( y  e.  ( B  \  |^|_ x  e.  A  C
)  <->  ( y  e.  B  /\  -.  y  e.  |^|_ x  e.  A  C ) )
1310, 11, 123bitr4i 277 . 2  |-  ( y  e.  U_ x  e.  A  ( B  \  C )  <->  y  e.  ( B  \  |^|_ x  e.  A  C )
)
1413eqriv 2463 1  |-  U_ x  e.  A  ( B  \  C )  =  ( B  \  |^|_ x  e.  A  C )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   E.wrex 2818   _Vcvv 3118    \ cdif 3478   U_ciun 4331   |^|_ciin 4332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2822  df-rex 2823  df-v 3120  df-dif 3484  df-iun 4333  df-iin 4334
This theorem is referenced by:  iuncld  19414  pnrmopn  19712  alexsublem  20412  bcth3  21638  iundifdifd  27252  iundifdif  27253
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