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Theorem elex22 2969
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 2507 . . . 4  |-  ( A  e.  B  ->  (
x  =  A  ->  x  e.  B )
)
2 eleq1a 2507 . . . 4  |-  ( A  e.  C  ->  (
x  =  A  ->  x  e.  C )
)
31, 2anim12ii 555 . . 3  |-  ( ( A  e.  B  /\  A  e.  C )  ->  ( x  =  A  ->  ( x  e.  B  /\  x  e.  C ) ) )
43alrimiv 1642 . 2  |-  ( ( A  e.  B  /\  A  e.  C )  ->  A. x ( x  =  A  ->  (
x  e.  B  /\  x  e.  C )
) )
5 elisset 2968 . . 3  |-  ( A  e.  B  ->  E. x  x  =  A )
65adantr 453 . 2  |-  ( ( A  e.  B  /\  A  e.  C )  ->  E. x  x  =  A )
7 exim 1585 . 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 59 1  |-  ( ( A  e.  B  /\  A  e.  C )  ->  E. x ( x  e.  B  /\  x  e.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360   A.wal 1550   E.wex 1551    = wceq 1653    e. wcel 1726
This theorem is referenced by:  en3lplem1VD  29017
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-11 1762  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-v 2960
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