Theorem en3lplem1VD 39078

Description: Virtual deduction proof of en3lplem1 8511. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
en3lplem1VD	|- ( ( A e. B /\ B e. C /\ C e. A ) -> ( x = A -> E. y ( y e. { A , B , C } /\ y e. x ) ) )

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

Proof of Theorem en3lplem1VD

Step	Hyp	Ref	Expression
1		idn1 38790	. . . . . . 7 |- (. ( A e. B /\ B e. C /\ C e. A ) ->. ( A e. B /\ B e. C /\ C e. A ) ).
2		simp3 1063	. . . . . . 7 |- ( ( A e. B /\ B e. C /\ C e. A ) -> C e. A )
3	1, 2	e1a 38852	. . . . . 6 |- (. ( A e. B /\ B e. C /\ C e. A ) ->. C e. A ).
4		tpid3g 4305	. . . . . 6 |- ( C e. A -> C e. { A , B , C } )
5	3, 4	e1a 38852	. . . . 5 |- (. ( A e. B /\ B e. C /\ C e. A ) ->. C e. { A , B , C } ).
6		idn2 38838	. . . . . 6 |- (. ( A e. B /\ B e. C /\ C e. A ) ,. x = A ->. x = A ).
7		eleq2 2690	. . . . . . 7 |- ( x = A -> ( C e. x <-> C e. A ) )
8	7	biimprd 238	. . . . . 6 |- ( x = A -> ( C e. A -> C e. x ) )
9	6, 3, 8	e21 38957	. . . . 5 |- (. ( A e. B /\ B e. C /\ C e. A ) ,. x = A ->. C e. x ).
10		pm3.2 463	. . . . 5 |- ( C e. { A , B , C } -> ( C e. x -> ( C e. { A , B , C } /\ C e. x ) ) )
11	5, 9, 10	e12 38951	. . . 4 |- (. ( A e. B /\ B e. C /\ C e. A ) ,. x = A ->. ( C e. { A , B , C } /\ C e. x ) ).
12		elex22 3217	. . . 4 |- ( ( C e. { A , B , C } /\ C e. x ) -> E. y ( y e. { A , B , C } /\ y e. x ) )
13	11, 12	e2 38856	. . 3 |- (. ( A e. B /\ B e. C /\ C e. A ) ,. x = A ->. E. y ( y e. { A , B , C } /\ y e. x ) ).
14	13	in2 38830	. 2 |- (. ( A e. B /\ B e. C /\ C e. A ) ->. ( x = A -> E. y ( y e. { A , B , C } /\ y e. x ) ) ).
15	14	in1 38787	1 |- ( ( A e. B /\ B e. C /\ C e. A ) -> ( x = A -> E. y ( y e. { A , B , C } /\ y e. x ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {ctp 4181

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-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-sn 4178  df-pr 4180  df-tp 4182  df-vd1 38786  df-vd2 38794

This theorem is referenced by:  en3lplem2VD  39079