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Theorem exists2 2233
 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 2153 . . . . . 6
2 nfa1 1756 . . . . . 6
3 exists1 2232 . . . . . . 7
4 ax16 1985 . . . . . . 7
53, 4sylbi 187 . . . . . 6
61, 2, 5exlimd 1803 . . . . 5
76com12 27 . . . 4
8 alex 1559 . . . 4
97, 8syl6ib 217 . . 3
109con2d 107 . 2
1110imp 418 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 358  wal 1527  wex 1528   wceq 1623  weu 2143 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-eu 2147
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