Theorem exists2 2562

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		nfeu1 2480	. . . . . 6 |- F/ x E! x x = x
2		nfa1 2028	. . . . . 6 |- F/ x A. x ph
3		exists1 2561	. . . . . . 7 |- ( E! x x = x <-> A. x x = y )
4		axc16 2135	. . . . . . 7 |- ( A. x x = y -> ( ph -> A. x ph ) )
5	3, 4	sylbi 207	. . . . . 6 |- ( E! x x = x -> ( ph -> A. x ph ) )
6	1, 2, 5	exlimd 2087	. . . . 5 |- ( E! x x = x -> ( E. x ph -> A. x ph ) )
7	6	com12 32	. . . 4 |- ( E. x ph -> ( E! x x = x -> A. x ph ) )
8		alex 1753	. . . 4 |- ( A. x ph <-> -. E. x -. ph )
9	7, 8	syl6ib 241	. . 3 |- ( E. x ph -> ( E! x x = x -> -. E. x -. ph ) )
10	9	con2d 129	. 2 |- ( E. x ph -> ( E. x -. ph -> -. E! x x = x ) )
11	10	imp 445	1 |- ( ( E. x ph /\ E. x -. ph ) -> -. E! x x = x )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   A.wal 1481   E.wex 1704   E!weu 2470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-eu 2474

This theorem is referenced by: (None)