Theorem elex2VD 39073

Description: Virtual deduction proof of elex2 3216. (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

Step	Hyp	Ref	Expression
1		idn1 38790	. . . . . 6 |- (. A e. B ->. A e. B ).
2		idn2 38838	. . . . . 6 |- (. A e. B ,. x = A ->. x = A ).
3		eleq1a 2696	. . . . . 6 |- ( A e. B -> ( x = A -> x e. B ) )
4	1, 2, 3	e12 38951	. . . . 5 |- (. A e. B ,. x = A ->. x e. B ).
5	4	in2 38830	. . . 4 |- (. A e. B ->. ( x = A -> x e. B ) ).
6	5	gen11 38841	. . 3 |- (. A e. B ->. A. x ( x = A -> x e. B ) ).
7		elisset 3215	. . . 4 |- ( A e. B -> E. x x = A )
8	1, 7	e1a 38852	. . 3 |- (. A e. B ->. E. x x = A ).
9		exim 1761	. . 3 |- ( A. x ( x = A -> x e. B ) -> ( E. x x = A -> E. x x e. B ) )
10	6, 8, 9	e11 38913	. 2 |- (. A e. B ->. E. x x e. B ).
11	10	in1 38787	1 |- ( A e. B -> E. x x e. B )

Syntax hints:   -> wi 4   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-tru 1486  df-ex 1705  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202  df-vd1 38786  df-vd2 38794

This theorem is referenced by: (None)