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Theorem elunif 30788
Description: A version of eluni 4241 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 4241 . 2  |-  ( A  e.  U. B  <->  E. y
( A  e.  y  /\  y  e.  B
) )
2 elunif.1 . . . . 5  |-  F/_ x A
3 nfcv 2622 . . . . 5  |-  F/_ x
y
42, 3nfel 2635 . . . 4  |-  F/ x  A  e.  y
5 elunif.2 . . . . 5  |-  F/_ x B
63, 5nfel 2635 . . . 4  |-  F/ x  y  e.  B
74, 6nfan 1870 . . 3  |-  F/ x
( A  e.  y  /\  y  e.  B
)
8 nfv 1678 . . 3  |-  F/ y ( A  e.  x  /\  x  e.  B
)
9 eleq2 2533 . . . 4  |-  ( y  =  x  ->  ( A  e.  y  <->  A  e.  x ) )
10 eleq1 2532 . . . 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 1988 . 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 1591    e. wcel 1762   F/_wnfc 2608   U.cuni 4238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-v 3108  df-uni 4239
This theorem is referenced by:  stoweidlem46  31165  stoweidlem57  31176
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