Theorem elex22VD 39074

Description: Virtual deduction proof of elex22 3217. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
elex22VD	|- ( ( A e. B /\ A e. C ) -> E. x ( x e. B /\ x e. C ) )

Distinct variable groups:   x, A   x, B   x, C

Proof of Theorem elex22VD

Step	Hyp	Ref	Expression
1		idn1 38790	. . . . 5 |- (. ( A e. B /\ A e. C ) ->. ( A e. B /\ A e. C ) ).
2		simpl 473	. . . . 5 |- ( ( A e. B /\ A e. C ) -> A e. B )
3	1, 2	e1a 38852	. . . 4 |- (. ( A e. B /\ A e. C ) ->. A e. B ).
4		elisset 3215	. . . 4 |- ( A e. B -> E. x x = A )
5	3, 4	e1a 38852	. . 3 |- (. ( A e. B /\ A e. C ) ->. E. x x = A ).
6		idn2 38838	. . . . . . . 8 |- (. ( A e. B /\ A e. C ) ,. x = A ->. x = A ).
7		eleq1a 2696	. . . . . . . 8 |- ( A e. B -> ( x = A -> x e. B ) )
8	3, 6, 7	e12 38951	. . . . . . 7 |- (. ( A e. B /\ A e. C ) ,. x = A ->. x e. B ).
9		simpr 477	. . . . . . . . 9 |- ( ( A e. B /\ A e. C ) -> A e. C )
10	1, 9	e1a 38852	. . . . . . . 8 |- (. ( A e. B /\ A e. C ) ->. A e. C ).
11		eleq1a 2696	. . . . . . . 8 |- ( A e. C -> ( x = A -> x e. C ) )
12	10, 6, 11	e12 38951	. . . . . . 7 |- (. ( A e. B /\ A e. C ) ,. x = A ->. x e. C ).
13		pm3.2 463	. . . . . . 7 |- ( x e. B -> ( x e. C -> ( x e. B /\ x e. C ) ) )
14	8, 12, 13	e22 38896	. . . . . 6 |- (. ( A e. B /\ A e. C ) ,. x = A ->. ( x e. B /\ x e. C ) ).
15	14	in2 38830	. . . . 5 |- (. ( A e. B /\ A e. C ) ->. ( x = A -> ( x e. B /\ x e. C ) ) ).
16	15	gen11 38841	. . . 4 |- (. ( A e. B /\ A e. C ) ->. A. x ( x = A -> ( x e. B /\ x e. C ) ) ).
17		exim 1761	. . . 4 |- ( A. x ( x = A -> ( x e. B /\ x e. C ) ) -> ( E. x x = A -> E. x ( x e. B /\ x e. C ) ) )
18	16, 17	e1a 38852	. . 3 |- (. ( A e. B /\ A e. C ) ->. ( E. x x = A -> E. x ( x e. B /\ x e. C ) ) ).
19		pm2.27 42	. . 3 |- ( E. x x = A -> ( ( E. x x = A -> E. x ( x e. B /\ x e. C ) ) -> E. x ( x e. B /\ x e. C ) ) )
20	5, 18, 19	e11 38913	. 2 |- (. ( A e. B /\ A e. C ) ->. E. x ( x e. B /\ x e. C ) ).
21	20	in1 38787	1 |- ( ( A e. B /\ A e. C ) -> E. x ( x e. B /\ x e. C ) )

Syntax hints:   -> wi 4   /\ wa 384   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)