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Theorem elex22 3071
Description: If two classes each contain another class, then both contain some set. (Contributed by Alan Sare, 24-Oct-2011.)
Assertion
Ref Expression
elex22  |-  ( ( A  e.  B  /\  A  e.  C )  ->  E. x ( x  e.  B  /\  x  e.  C ) )
Distinct variable groups:    x, A    x, B    x, C

Proof of Theorem elex22
StepHypRef Expression
1 eleq1a 2535 . . . 4  |-  ( A  e.  B  ->  (
x  =  A  ->  x  e.  B )
)
2 eleq1a 2535 . . . 4  |-  ( A  e.  C  ->  (
x  =  A  ->  x  e.  C )
)
31, 2anim12ii 578 . . 3  |-  ( ( A  e.  B  /\  A  e.  C )  ->  ( x  =  A  ->  ( x  e.  B  /\  x  e.  C ) ) )
43alrimiv 1784 . 2  |-  ( ( A  e.  B  /\  A  e.  C )  ->  A. x ( x  =  A  ->  (
x  e.  B  /\  x  e.  C )
) )
5 elisset 3069 . . 3  |-  ( A  e.  B  ->  E. x  x  =  A )
65adantr 471 . 2  |-  ( ( A  e.  B  /\  A  e.  C )  ->  E. x  x  =  A )
7 exim 1717 . 2  |-  ( A. x ( x  =  A  ->  ( x  e.  B  /\  x  e.  C ) )  -> 
( E. x  x  =  A  ->  E. x
( x  e.  B  /\  x  e.  C
) ) )
84, 6, 7sylc 62 1  |-  ( ( A  e.  B  /\  A  e.  C )  ->  E. x ( x  e.  B  /\  x  e.  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375   A.wal 1453    = wceq 1455   E.wex 1674    e. wcel 1898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-12 1944  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1458  df-ex 1675  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-v 3059
This theorem is referenced by:  en3lplem1VD  37279
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