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

Step	Hyp	Ref	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 ) )
4	1, 2, 3	e12 33002	. . . . 5 |- (. A e. B ,. x = A ->. x e. B ).
5	4	in2 32872	. . . 4 |- (. A e. B ->. ( x = A -> x e. B ) ).
6	5	gen11 32883	. . 3 |- (. A e. B ->. A. x ( x = A -> x e. B ) ).
7		elisset 3129	. . . 4 |- ( A e. B -> E. x x = A )
8	1, 7	e1a 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 ) )
10	6, 8, 9	e11 32955	. 2 |- (. A e. B ->. E. x x e. B ).
11	10	in1 32829	1 |- ( A e. B -> E. x x e. B )

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)