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Theorem elex2VD 37228
Description: Virtual deduction proof of elex2 3057. (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 36938 . . . . . 6  |-  (. A  e.  B  ->.  A  e.  B ).
2 idn2 36986 . . . . . 6  |-  (. A  e.  B ,. x  =  A  ->.  x  =  A ).
3 eleq1a 2523 . . . . . 6  |-  ( A  e.  B  ->  (
x  =  A  ->  x  e.  B )
)
41, 2, 3e12 37105 . . . . 5  |-  (. A  e.  B ,. x  =  A  ->.  x  e.  B ).
54in2 36978 . . . 4  |-  (. A  e.  B  ->.  ( x  =  A  ->  x  e.  B ) ).
65gen11 36989 . . 3  |-  (. A  e.  B  ->.  A. x ( x  =  A  ->  x  e.  B ) ).
7 elisset 3056 . . . 4  |-  ( A  e.  B  ->  E. x  x  =  A )
81, 7e1a 37000 . . 3  |-  (. A  e.  B  ->.  E. x  x  =  A ).
9 exim 1705 . . 3  |-  ( A. x ( x  =  A  ->  x  e.  B )  ->  ( E. x  x  =  A  ->  E. x  x  e.  B ) )
106, 8, 9e11 37061 . 2  |-  (. A  e.  B  ->.  E. x  x  e.  B ).
1110in1 36935 1  |-  ( A  e.  B  ->  E. x  x  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1441    = wceq 1443   E.wex 1662    e. wcel 1886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-12 1932  ax-ext 2430
This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1446  df-ex 1663  df-sb 1797  df-clab 2437  df-cleq 2443  df-clel 2446  df-v 3046  df-vd1 36934  df-vd2 36942
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator