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Theorem elex2VD 29012
Description: Virtual deduction proof of elex2 2970. (Contributed by Alan Sare, 25-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
elex2VD  |-  ( A  e.  B  ->  E. x  x  e.  B )
Distinct variable groups:    x, A    x, B

Proof of Theorem elex2VD
StepHypRef Expression
1 idn1 28727 . . . . . 6  |-  (. A  e.  B  ->.  A  e.  B ).
2 idn2 28776 . . . . . 6  |-  (. A  e.  B ,. x  =  A  ->.  x  =  A ).
3 eleq1a 2507 . . . . . 6  |-  ( A  e.  B  ->  (
x  =  A  ->  x  e.  B )
)
41, 2, 3e12 28898 . . . . 5  |-  (. A  e.  B ,. x  =  A  ->.  x  e.  B ).
54in2 28768 . . . 4  |-  (. A  e.  B  ->.  ( x  =  A  ->  x  e.  B ) ).
65gen11 28779 . . 3  |-  (. A  e.  B  ->.  A. x ( x  =  A  ->  x  e.  B ) ).
7 elisset 2968 . . . 4  |-  ( A  e.  B  ->  E. x  x  =  A )
81, 7e1_ 28790 . . 3  |-  (. A  e.  B  ->.  E. x  x  =  A ).
9 exim 1585 . . 3  |-  ( A. x ( x  =  A  ->  x  e.  B )  ->  ( E. x  x  =  A  ->  E. x  x  e.  B ) )
106, 8, 9e11 28851 . 2  |-  (. A  e.  B  ->.  E. x  x  e.  B ).
1110in1 28724 1  |-  ( A  e.  B  ->  E. x  x  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1550   E.wex 1551    = wceq 1653    e. wcel 1726
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  df-vd1 28723  df-vd2 28732
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