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Theorem nrex 2501
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 2462 . 2 𝑥𝐴 ¬ 𝜓
3 ralnex 2403 . 2 (∀𝑥𝐴 ¬ 𝜓 ↔ ¬ ∃𝑥𝐴 𝜓)
42, 3mpbi 144 1 ¬ ∃𝑥𝐴 𝜓
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wcel 1465  wral 2393  wrex 2394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-5 1408  ax-gen 1410  ax-ie2 1455
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-fal 1322  df-ral 2398  df-rex 2399
This theorem is referenced by:  rex0  3350  iun0  3839  frec0g  6262  nominpos  8925  sqrt2irr  11767  exmidsbthrlem  13144
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