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Theorem elex2VD 33119
Description: Virtual deduction proof of elex2 3130. (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 32832 . . . . . 6  |-  (. A  e.  B  ->.  A  e.  B ).
2 idn2 32880 . . . . . 6  |-  (. A  e.  B ,. x  =  A  ->.  x  =  A ).
3 eleq1a 2550 . . . . . 6  |-  ( A  e.  B  ->  (
x  =  A  ->  x  e.  B )
)
41, 2, 3e12 33002 . . . . 5  |-  (. A  e.  B ,. x  =  A  ->.  x  e.  B ).
54in2 32872 . . . 4  |-  (. A  e.  B  ->.  ( x  =  A  ->  x  e.  B ) ).
65gen11 32883 . . 3  |-  (. A  e.  B  ->.  A. x ( x  =  A  ->  x  e.  B ) ).
7 elisset 3129 . . . 4  |-  ( A  e.  B  ->  E. x  x  =  A )
81, 7e1a 32894 . . 3  |-  (. A  e.  B  ->.  E. x  x  =  A ).
9 exim 1633 . . 3  |-  ( A. x ( x  =  A  ->  x  e.  B )  ->  ( E. x  x  =  A  ->  E. x  x  e.  B ) )
106, 8, 9e11 32955 . 2  |-  (. A  e.  B  ->.  E. x  x  e.  B ).
1110in1 32829 1  |-  ( A  e.  B  ->  E. x  x  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1377    = wceq 1379   E.wex 1596    e. wcel 1767
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-12 1803  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-v 3120  df-vd1 32828  df-vd2 32836
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator