Theorem exists2 2045

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 1958	. . . . . 6 |- ( E! x x = x -> A. x E! x x = x )
2		hba1 1478	. . . . . 6 |- ( A. x ph -> A. x A. x ph )
3		exists1 2044	. . . . . . 7 |- ( E! x x = x <-> A. x x = y )
4		ax16 1741	. . . . . . 7 |- ( A. x x = y -> ( ph -> A. x ph ) )
5	3, 4	sylbi 119	. . . . . 6 |- ( E! x x = x -> ( ph -> A. x ph ) )
6	1, 2, 5	exlimdh 1532	. . . . 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 1581	. . . 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 589	. 2 |- ( E. x ph -> ( E. x -. ph -> -. E! x x = x ) )
11	10	imp 122	1 |- ( ( E. x ph /\ E. x -. ph ) -> -. E! x x = x )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   A.wal 1287   = wceq 1289   E.wex 1426   E!weu 1948

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951

This theorem is referenced by: (None)