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

Step	Hyp	Ref	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 ) )
4	1, 2, 3	e12 31291	. . . . 5 |- (. A e. B ,. x = A ->. x e. B ).
5	4	in2 31161	. . . 4 |- (. A e. B ->. ( x = A -> x e. B ) ).
6	5	gen11 31172	. . 3 |- (. A e. B ->. A. x ( x = A -> x e. B ) ).
7		elisset 2981	. . . 4 |- ( A e. B -> E. x x = A )
8	1, 7	e1_ 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 ) )
10	6, 8, 9	e11 31244	. 2 |- (. A e. B ->. E. x x e. B ).
11	10	in1 31117	1 |- ( A e. B -> E. x x e. B )

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)