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Theorem rex0 3563
 Description: Vacuous existential quantification is false. (Contributed by NM, 15-Oct-2003.)
Assertion
Ref Expression
rex0 ¬ x φ

Proof of Theorem rex0
StepHypRef Expression
1 noel 3554 . . 3 ¬ x
21pm2.21i 123 . 2 (x → ¬ φ)
32nrex 2716 1 ¬ x φ
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∈ wcel 1710  ∃wrex 2615  ∅c0 3550 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  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551 This theorem is referenced by:  0iun  4023  addcnul1  4452  nnadjoin  4520  tfinnn  4534
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