Theorem exists2 2175

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 2087	. . . . . 6 |- ( E! x x = x -> A. x E! x x = x )
2		hba1 1586	. . . . . 6 |- ( A. x ph -> A. x A. x ph )
3		exists1 2174	. . . . . . 7 |- ( E! x x = x <-> A. x x = y )
4		ax16 1859	. . . . . . 7 |- ( A. x x = y -> ( ph -> A. x ph ) )
5	3, 4	sylbi 121	. . . . . 6 |- ( E! x x = x -> ( ph -> A. x ph ) )
6	1, 2, 5	exlimdh 1642	. . . . 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 1691	. . . 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 627	. 2 |- ( E. x ph -> ( E. x -. ph -> -. E! x x = x ) )
11	10	imp 124	1 |- ( ( E. x ph /\ E. x -. ph ) -> -. E! x x = x )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   A.wal 1393   = wceq 1395   E.wex 1538   E!weu 2077

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080

This theorem is referenced by: (None)