Theorem exists2 1997

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 1910	. . . . . 6 |- ( E! x x = x -> A. x E! x x = x )
2		hba1 1433	. . . . . 6 |- ( A. x ph -> A. x A. x ph )
3		exists1 1996	. . . . . . 7 |- ( E! x x = x <-> A. x x = y )
4		ax16 1694	. . . . . . 7 |- ( A. x x = y -> ( ph -> A. x ph ) )
5	3, 4	sylbi 114	. . . . . 6 |- ( E! x x = x -> ( ph -> A. x ph ) )
6	1, 2, 5	exlimdh 1487	. . . . 5 |- ( E! x x = x -> ( E. x ph -> A. x ph ) )
7	6	com12 27	. . . 4 |- ( E. x ph -> ( E! x x = x -> A. x ph ) )
8		alexim 1536	. . . 4 |- ( A. x ph -> -. E. x -. ph )
9	7, 8	syl6 29	. . 3 |- ( E. x ph -> ( E! x x = x -> -. E. x -. ph ) )
10	9	con2d 554	. 2 |- ( E. x ph -> ( E. x -. ph -> -. E! x x = x ) )
11	10	imp 115	1 |- ( ( E. x ph /\ E. x -. ph ) -> -. E! x x = x )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   A.wal 1241   = wceq 1243   E.wex 1381   E!weu 1900

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903

This theorem is referenced by: (None)