Theorem exists2 2294

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.xph /\ 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 2214	. . . . . 6 |- F/xE!x x = x
2		nfa1 1788	. . . . . 6 |- F/xA.xph
3		exists1 2293	. . . . . . 7 |- (E!x x = x <-> A.x x = y)
4		ax16 2045	. . . . . . 7 |- (A.x x = y -> (ph -> A.xph))
5	3, 4	sylbi 187	. . . . . 6 |- (E!x x = x -> (ph -> A.xph))
6	1, 2, 5	exlimd 1806	. . . . 5 |- (E!x x = x -> (E.xph -> A.xph))
7	6	com12 27	. . . 4 |- (E.xph -> (E!x x = x -> A.xph))
8		alex 1572	. . . 4 |- (A.xph <-> -. E.x -. ph)
9	7, 8	syl6ib 217	. . 3 |- (E.xph -> (E!x x = x -> -. E.x -. ph))
10	9	con2d 107	. 2 |- (E.xph -> (E.x -. ph -> -. E!x x = x))
11	10	imp 418	1 |- ((E.xph /\ E.x -. ph) -> -. E!x x = x)

Syntax hints:  -. wn 3   -> wi 4   /\ wa 358  A.wal 1540  E.wex 1541   = wceq 1642  E!weu 2204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-eu 2208

This theorem is referenced by: (None)