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Theorem elex22VD 28292
Description: Virtual deduction proof of elex22 2910. (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 28006 . . . . 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_ 28069 . . . 4  |-  (. ( A  e.  B  /\  A  e.  C )  ->.  A  e.  B ).
4 elisset 2909 . . . 4  |-  ( A  e.  B  ->  E. x  x  =  A )
53, 4e1_ 28069 . . 3  |-  (. ( A  e.  B  /\  A  e.  C )  ->.  E. x  x  =  A ).
6 idn2 28055 . . . . . . . 8  |-  (. ( A  e.  B  /\  A  e.  C ) ,. x  =  A  ->.  x  =  A ).
7 eleq1a 2456 . . . . . . . 8  |-  ( A  e.  B  ->  (
x  =  A  ->  x  e.  B )
)
83, 6, 7e12 28177 . . . . . . 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_ 28069 . . . . . . . 8  |-  (. ( A  e.  B  /\  A  e.  C )  ->.  A  e.  C ).
11 eleq1a 2456 . . . . . . . 8  |-  ( A  e.  C  ->  (
x  =  A  ->  x  e.  C )
)
1210, 6, 11e12 28177 . . . . . . 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 28113 . . . . . 6  |-  (. ( A  e.  B  /\  A  e.  C ) ,. x  =  A  ->.  ( x  e.  B  /\  x  e.  C ) ).
1514in2 28047 . . . . 5  |-  (. ( A  e.  B  /\  A  e.  C )  ->.  ( x  =  A  -> 
( x  e.  B  /\  x  e.  C
) ) ).
1615gen11 28058 . . . 4  |-  (. ( A  e.  B  /\  A  e.  C )  ->.  A. x ( x  =  A  ->  ( x  e.  B  /\  x  e.  C ) ) ).
17 exim 1581 . . . 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_ 28069 . . 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 28130 . 2  |-  (. ( A  e.  B  /\  A  e.  C )  ->.  E. x ( x  e.  B  /\  x  e.  C ) ).
2120in1 28003 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 1546   E.wex 1547    = wceq 1649    e. wcel 1717
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-11 1753  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-v 2901  df-vd1 28002  df-vd2 28011
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