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Theorem elex22VD 28888
Description: Virtual deduction proof of elex22 2959. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
elex22VD  |-  ( ( 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 elex22VD
StepHypRef Expression
1 idn1 28602 . . . . 5  |-  (. ( A  e.  B  /\  A  e.  C )  ->.  ( A  e.  B  /\  A  e.  C ) ).
2 simpl 444 . . . . 5  |-  ( ( A  e.  B  /\  A  e.  C )  ->  A  e.  B )
31, 2e1_ 28665 . . . 4  |-  (. ( A  e.  B  /\  A  e.  C )  ->.  A  e.  B ).
4 elisset 2958 . . . 4  |-  ( A  e.  B  ->  E. x  x  =  A )
53, 4e1_ 28665 . . 3  |-  (. ( A  e.  B  /\  A  e.  C )  ->.  E. x  x  =  A ).
6 idn2 28651 . . . . . . . 8  |-  (. ( A  e.  B  /\  A  e.  C ) ,. x  =  A  ->.  x  =  A ).
7 eleq1a 2504 . . . . . . . 8  |-  ( A  e.  B  ->  (
x  =  A  ->  x  e.  B )
)
83, 6, 7e12 28773 . . . . . . 7  |-  (. ( A  e.  B  /\  A  e.  C ) ,. x  =  A  ->.  x  e.  B ).
9 simpr 448 . . . . . . . . 9  |-  ( ( A  e.  B  /\  A  e.  C )  ->  A  e.  C )
101, 9e1_ 28665 . . . . . . . 8  |-  (. ( A  e.  B  /\  A  e.  C )  ->.  A  e.  C ).
11 eleq1a 2504 . . . . . . . 8  |-  ( A  e.  C  ->  (
x  =  A  ->  x  e.  C )
)
1210, 6, 11e12 28773 . . . . . . 7  |-  (. ( A  e.  B  /\  A  e.  C ) ,. x  =  A  ->.  x  e.  C ).
13 pm3.2 435 . . . . . . 7  |-  ( x  e.  B  ->  (
x  e.  C  -> 
( x  e.  B  /\  x  e.  C
) ) )
148, 12, 13e22 28709 . . . . . 6  |-  (. ( A  e.  B  /\  A  e.  C ) ,. x  =  A  ->.  ( x  e.  B  /\  x  e.  C ) ).
1514in2 28643 . . . . 5  |-  (. ( A  e.  B  /\  A  e.  C )  ->.  ( x  =  A  -> 
( x  e.  B  /\  x  e.  C
) ) ).
1615gen11 28654 . . . 4  |-  (. ( A  e.  B  /\  A  e.  C )  ->.  A. x ( x  =  A  ->  ( x  e.  B  /\  x  e.  C ) ) ).
17 exim 1584 . . . 4  |-  ( A. x ( x  =  A  ->  ( x  e.  B  /\  x  e.  C ) )  -> 
( E. x  x  =  A  ->  E. x
( x  e.  B  /\  x  e.  C
) ) )
1816, 17e1_ 28665 . . 3  |-  (. ( A  e.  B  /\  A  e.  C )  ->.  ( E. x  x  =  A  ->  E. x
( x  e.  B  /\  x  e.  C
) ) ).
19 pm2.27 37 . . 3  |-  ( E. x  x  =  A  ->  ( ( E. x  x  =  A  ->  E. x ( x  e.  B  /\  x  e.  C ) )  ->  E. x ( x  e.  B  /\  x  e.  C ) ) )
205, 18, 19e11 28726 . 2  |-  (. ( A  e.  B  /\  A  e.  C )  ->.  E. x ( x  e.  B  /\  x  e.  C ) ).
2120in1 28599 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 359   A.wal 1549   E.wex 1550    = wceq 1652    e. wcel 1725
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-11 1761  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-v 2950  df-vd1 28598  df-vd2 28607
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