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Theorem elex22 2812
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 2365 . . . 4  |-  ( A  e.  B  ->  (
x  =  A  ->  x  e.  B )
)
2 eleq1a 2365 . . . 4  |-  ( A  e.  C  ->  (
x  =  A  ->  x  e.  C )
)
31, 2anim12ii 553 . . 3  |-  ( ( A  e.  B  /\  A  e.  C )  ->  ( x  =  A  ->  ( x  e.  B  /\  x  e.  C ) ) )
43alrimiv 1621 . 2  |-  ( ( A  e.  B  /\  A  e.  C )  ->  A. x ( x  =  A  ->  (
x  e.  B  /\  x  e.  C )
) )
5 elisset 2811 . . 3  |-  ( A  e.  B  ->  E. x  x  =  A )
65adantr 451 . 2  |-  ( ( A  e.  B  /\  A  e.  C )  ->  E. x  x  =  A )
7 exim 1565 . 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 56 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 358   A.wal 1530   E.wex 1531    = wceq 1632    e. wcel 1696
This theorem is referenced by:  en3lplem1VD  28935
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-11 1727  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-v 2803
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