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Theorem nrex 2428
 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 2391 . 2 𝑥𝐴 ¬ 𝜓
3 ralnex 2333 . 2 (∀𝑥𝐴 ¬ 𝜓 ↔ ¬ ∃𝑥𝐴 𝜓)
42, 3mpbi 137 1 ¬ ∃𝑥𝐴 𝜓
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 1409  ∀wral 2323  ∃wrex 2324 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-5 1352  ax-gen 1354  ax-ie2 1399 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-ral 2328  df-rex 2329 This theorem is referenced by:  rex0  3266  iun0  3741  frec0g  6014  nominpos  8219  sqrt2irr  10251
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