Theorem exists2 2094

Description: A condition implying that at least two things exist. (Contributed by NM, 10-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Assertion

Ref	Expression
exists2	|- ( ( E. x ph /\ E. x -. ph ) -> -. E! x x = x )

Proof of Theorem exists2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hbeu1 2007	. . . . . 6 |- ( E! x x = x -> A. x E! x x = x )
2		hba1 1520	. . . . . 6 |- ( A. x ph -> A. x A. x ph )
3		exists1 2093	. . . . . . 7 |- ( E! x x = x <-> A. x x = y )
4		ax16 1785	. . . . . . 7 |- ( A. x x = y -> ( ph -> A. x ph ) )
5	3, 4	sylbi 120	. . . . . 6 |- ( E! x x = x -> ( ph -> A. x ph ) )
6	1, 2, 5	exlimdh 1575	. . . . 5 |- ( E! x x = x -> ( E. x ph -> A. x ph ) )
7	6	com12 30	. . . 4 |- ( E. x ph -> ( E! x x = x -> A. x ph ) )
8		alexim 1624	. . . 4 |- ( A. x ph -> -. E. x -. ph )
9	7, 8	syl6 33	. . 3 |- ( E. x ph -> ( E! x x = x -> -. E. x -. ph ) )
10	9	con2d 613	. 2 |- ( E. x ph -> ( E. x -. ph -> -. E! x x = x ) )
11	10	imp 123	1 |- ( ( E. x ph /\ E. x -. ph ) -> -. E! x x = x )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   A.wal 1329   = wceq 1331   E.wex 1468   E!weu 1997

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000

This theorem is referenced by: (None)