Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elunif Structured version   Unicode version

Theorem elunif 29583
Description: A version of eluni 4082 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 4082 . 2  |-  ( A  e.  U. B  <->  E. y
( A  e.  y  /\  y  e.  B
) )
2 elunif.1 . . . . 5  |-  F/_ x A
3 nfcv 2569 . . . . 5  |-  F/_ x
y
42, 3nfel 2577 . . . 4  |-  F/ x  A  e.  y
5 elunif.2 . . . . 5  |-  F/_ x B
63, 5nfel 2577 . . . 4  |-  F/ x  y  e.  B
74, 6nfan 1859 . . 3  |-  F/ x
( A  e.  y  /\  y  e.  B
)
8 nfv 1672 . . 3  |-  F/ y ( A  e.  x  /\  x  e.  B
)
9 eleq2 2494 . . . 4  |-  ( y  =  x  ->  ( A  e.  y  <->  A  e.  x ) )
10 eleq1 2493 . . . 4  |-  ( y  =  x  ->  (
y  e.  B  <->  x  e.  B ) )
119, 10anbi12d 703 . . 3  |-  ( y  =  x  ->  (
( A  e.  y  /\  y  e.  B
)  <->  ( A  e.  x  /\  x  e.  B ) ) )
127, 8, 11cbvex 1969 . 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 1589    e. wcel 1755   F/_wnfc 2556   U.cuni 4079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-v 2964  df-uni 4080
This theorem is referenced by:  stoweidlem46  29687  stoweidlem57  29698
  Copyright terms: Public domain W3C validator