Theorem en3lplem1VD 33744

Description: Virtual deduction proof of en3lplem1 8048. (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 33452	. . . . . . 7 |- (. ( A e. B /\ B e. C /\ C e. A ) ->. ( A e. B /\ B e. C /\ C e. A ) ).
2		simp3 998	. . . . . . 7 |- ( ( A e. B /\ B e. C /\ C e. A ) -> C e. A )
3	1, 2	e1a 33514	. . . . . 6 |- (. ( A e. B /\ B e. C /\ C e. A ) ->. C e. A ).
4		tpid3g 4147	. . . . . 6 |- ( C e. A -> C e. { A , B , C } )
5	3, 4	e1a 33514	. . . . 5 |- (. ( A e. B /\ B e. C /\ C e. A ) ->. C e. { A , B , C } ).
6		idn2 33500	. . . . . 6 |- (. ( A e. B /\ B e. C /\ C e. A ) ,. x = A ->. x = A ).
7		eleq2 2530	. . . . . . 7 |- ( x = A -> ( C e. x <-> C e. A ) )
8	7	biimprd 223	. . . . . 6 |- ( x = A -> ( C e. A -> C e. x ) )
9	6, 3, 8	e21 33628	. . . . 5 |- (. ( A e. B /\ B e. C /\ C e. A ) ,. x = A ->. C e. x ).
10		pm3.2 447	. . . . 5 |- ( C e. { A , B , C } -> ( C e. x -> ( C e. { A , B , C } /\ C e. x ) ) )
11	5, 9, 10	e12 33622	. . . 4 |- (. ( A e. B /\ B e. C /\ C e. A ) ,. x = A ->. ( C e. { A , B , C } /\ C e. x ) ).
12		elex22 3122	. . . 4 |- ( ( C e. { A , B , C } /\ C e. x ) -> E. y ( y e. { A , B , C } /\ y e. x ) )
13	11, 12	e2 33518	. . 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 33492	. 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 33449	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 369   /\ w3a 973   = wceq 1395   E.wex 1613   e. wcel 1819   {ctp 4036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-un 3476  df-sn 4033  df-pr 4035  df-tp 4037  df-vd1 33448  df-vd2 33456

This theorem is referenced by:  en3lplem2VD  33745