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Theorem nrex 2465
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 2428 . 2 𝑥𝐴 ¬ 𝜓
3 ralnex 2369 . 2 (∀𝑥𝐴 ¬ 𝜓 ↔ ¬ ∃𝑥𝐴 𝜓)
42, 3mpbi 143 1 ¬ ∃𝑥𝐴 𝜓
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wcel 1438  wral 2359  wrex 2360
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-5 1381  ax-gen 1383  ax-ie2 1428
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-ral 2364  df-rex 2365
This theorem is referenced by:  rex0  3298  iun0  3781  frec0g  6144  nominpos  8623  sqrt2irr  11221  exmidsbthrlem  11556
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