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Theorem exists2 2372
 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 2292 . . . . . 6
2 nfa1 1807 . . . . . 6
3 exists1 2371 . . . . . . 7
4 ax16 2051 . . . . . . 7
53, 4sylbi 189 . . . . . 6
61, 2, 5exlimd 1825 . . . . 5
76com12 30 . . . 4
8 alex 1582 . . . 4
97, 8syl6ib 219 . . 3
109con2d 110 . 2
1110imp 420 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 360  wal 1550  wex 1551  weu 2282 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951 This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-nf 1555  df-eu 2286
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