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Theorem elunif 27664
Description: A version of eluni 4019 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 4019 . 2  |-  ( A  e.  U. B  <->  E. y
( A  e.  y  /\  y  e.  B
) )
2 elunif.1 . . . . 5  |-  F/_ x A
3 nfcv 2573 . . . . 5  |-  F/_ x
y
42, 3nfel 2581 . . . 4  |-  F/ x  A  e.  y
5 elunif.2 . . . . 5  |-  F/_ x B
63, 5nfel 2581 . . . 4  |-  F/ x  y  e.  B
74, 6nfan 1847 . . 3  |-  F/ x
( A  e.  y  /\  y  e.  B
)
8 nfv 1630 . . 3  |-  F/ y ( A  e.  x  /\  x  e.  B
)
9 eleq2 2498 . . . 4  |-  ( y  =  x  ->  ( A  e.  y  <->  A  e.  x ) )
10 eleq1 2497 . . . 4  |-  ( y  =  x  ->  (
y  e.  B  <->  x  e.  B ) )
119, 10anbi12d 693 . . 3  |-  ( y  =  x  ->  (
( A  e.  y  /\  y  e.  B
)  <->  ( A  e.  x  /\  x  e.  B ) ) )
127, 8, 11cbvex 1984 . 2  |-  ( E. y ( A  e.  y  /\  y  e.  B )  <->  E. x
( A  e.  x  /\  x  e.  B
) )
131, 12bitri 242 1  |-  ( A  e.  U. B  <->  E. x
( A  e.  x  /\  x  e.  B
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360   E.wex 1551    e. wcel 1726   F/_wnfc 2560   U.cuni 4016
This theorem is referenced by:  stoweidlem46  27772  stoweidlem57  27783
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2418
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 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-v 2959  df-uni 4017
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