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Theorem elex2VD 31408
Description: Virtual deduction proof of elex2 2982. (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 31120 . . . . . 6  |-  (. A  e.  B  ->.  A  e.  B ).
2 idn2 31169 . . . . . 6  |-  (. A  e.  B ,. x  =  A  ->.  x  =  A ).
3 eleq1a 2510 . . . . . 6  |-  ( A  e.  B  ->  (
x  =  A  ->  x  e.  B )
)
41, 2, 3e12 31291 . . . . 5  |-  (. A  e.  B ,. x  =  A  ->.  x  e.  B ).
54in2 31161 . . . 4  |-  (. A  e.  B  ->.  ( x  =  A  ->  x  e.  B ) ).
65gen11 31172 . . 3  |-  (. A  e.  B  ->.  A. x ( x  =  A  ->  x  e.  B ) ).
7 elisset 2981 . . . 4  |-  ( A  e.  B  ->  E. x  x  =  A )
81, 7e1_ 31183 . . 3  |-  (. A  e.  B  ->.  E. x  x  =  A ).
9 exim 1628 . . 3  |-  ( A. x ( x  =  A  ->  x  e.  B )  ->  ( E. x  x  =  A  ->  E. x  x  e.  B ) )
106, 8, 9e11 31244 . 2  |-  (. A  e.  B  ->.  E. x  x  e.  B ).
1110in1 31117 1  |-  ( A  e.  B  ->  E. x  x  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1362    = wceq 1364   E.wex 1591    e. wcel 1761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-12 1797  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1592  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-v 2972  df-vd1 31116  df-vd2 31125
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator