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Theorem elunif 29878
Description: A version of eluni 4194 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
elunif.1  |-  F/_ x A
elunif.2  |-  F/_ x B
Assertion
Ref Expression
elunif  |-  ( A  e.  U. B  <->  E. x
( A  e.  x  /\  x  e.  B
) )
Distinct variable group:    A, B
Allowed substitution hints:    A( x)    B( x)

Proof of Theorem elunif
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eluni 4194 . 2  |-  ( A  e.  U. B  <->  E. y
( A  e.  y  /\  y  e.  B
) )
2 elunif.1 . . . . 5  |-  F/_ x A
3 nfcv 2613 . . . . 5  |-  F/_ x
y
42, 3nfel 2625 . . . 4  |-  F/ x  A  e.  y
5 elunif.2 . . . . 5  |-  F/_ x B
63, 5nfel 2625 . . . 4  |-  F/ x  y  e.  B
74, 6nfan 1863 . . 3  |-  F/ x
( A  e.  y  /\  y  e.  B
)
8 nfv 1674 . . 3  |-  F/ y ( A  e.  x  /\  x  e.  B
)
9 eleq2 2524 . . . 4  |-  ( y  =  x  ->  ( A  e.  y  <->  A  e.  x ) )
10 eleq1 2523 . . . 4  |-  ( y  =  x  ->  (
y  e.  B  <->  x  e.  B ) )
119, 10anbi12d 710 . . 3  |-  ( y  =  x  ->  (
( A  e.  y  /\  y  e.  B
)  <->  ( A  e.  x  /\  x  e.  B ) ) )
127, 8, 11cbvex 1979 . 2  |-  ( E. y ( A  e.  y  /\  y  e.  B )  <->  E. x
( A  e.  x  /\  x  e.  B
) )
131, 12bitri 249 1  |-  ( A  e.  U. B  <->  E. x
( A  e.  x  /\  x  e.  B
) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369   E.wex 1587    e. wcel 1758   F/_wnfc 2599   U.cuni 4191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-v 3072  df-uni 4192
This theorem is referenced by:  stoweidlem46  29981  stoweidlem57  29992
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