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Theorem nrex 2586
Description: Inference adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)
Hypothesis
Ref Expression
nrex.1 (𝑥𝐴 → ¬ 𝜓)
Assertion
Ref Expression
nrex ¬ ∃𝑥𝐴 𝜓

Proof of Theorem nrex
StepHypRef Expression
1 nrex.1 . . 3 (𝑥𝐴 → ¬ 𝜓)
21rgen 2547 . 2 𝑥𝐴 ¬ 𝜓
3 ralnex 2482 . 2 (∀𝑥𝐴 ¬ 𝜓 ↔ ¬ ∃𝑥𝐴 𝜓)
42, 3mpbi 145 1 ¬ ∃𝑥𝐴 𝜓
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wcel 2164  wral 2472  wrex 2473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1458  ax-gen 1460  ax-ie2 1505
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-ral 2477  df-rex 2478
This theorem is referenced by:  rex0  3464  iun0  3969  canth  5863  frec0g  6441  nominpos  9210  sqrt2irr  12290  gsum0g  12969  exmidsbthrlem  15457
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