Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  exists2 Structured version   Unicode version

Theorem exists2 2361
 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

Proof of Theorem exists2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfeu1 2279 . . . . . 6
2 nfa1 1956 . . . . . 6
3 exists1 2360 . . . . . . 7
4 axc16 2001 . . . . . . 7
53, 4sylbi 198 . . . . . 6
61, 2, 5exlimd 1974 . . . . 5
76com12 32 . . . 4
8 alex 1692 . . . 4
97, 8syl6ib 229 . . 3
109con2d 118 . 2
1110imp 430 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wa 370  wal 1435  wex 1657  weu 2269 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057 This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1658  df-nf 1662  df-eu 2273 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator