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Theorem elunif 27687
Description: A version of eluni 3830 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 3830 . 2  |-  ( A  e.  U. B  <->  E. y
( A  e.  y  /\  y  e.  B
) )
2 elunif.1 . . . . 5  |-  F/_ x A
3 nfcv 2419 . . . . 5  |-  F/_ x
y
42, 3nfel 2427 . . . 4  |-  F/ x  A  e.  y
5 elunif.2 . . . . 5  |-  F/_ x B
63, 5nfel 2427 . . . 4  |-  F/ x  y  e.  B
74, 6nfan 1771 . . 3  |-  F/ x
( A  e.  y  /\  y  e.  B
)
8 nfv 1605 . . 3  |-  F/ y ( A  e.  x  /\  x  e.  B
)
9 eleq2 2344 . . . 4  |-  ( y  =  x  ->  ( A  e.  y  <->  A  e.  x ) )
10 eleq1 2343 . . . 4  |-  ( y  =  x  ->  (
y  e.  B  <->  x  e.  B ) )
119, 10anbi12d 691 . . 3  |-  ( y  =  x  ->  (
( A  e.  y  /\  y  e.  B
)  <->  ( A  e.  x  /\  x  e.  B ) ) )
127, 8, 11cbvex 1925 . 2  |-  ( E. y ( A  e.  y  /\  y  e.  B )  <->  E. x
( A  e.  x  /\  x  e.  B
) )
131, 12bitri 240 1  |-  ( A  e.  U. B  <->  E. x
( A  e.  x  /\  x  e.  B
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   F/_wnfc 2406   U.cuni 3827
This theorem is referenced by:  stoweidlem46  27795  stoweidlem57  27806
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-uni 3828
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