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Theorem nexd 1575
Description: Deduction for generalization rule for negated wff. (Contributed by NM, 2-Jan-2002.)
Hypotheses
Ref Expression
nexd.1 (𝜑 → ∀𝑥𝜑)
nexd.2 (𝜑 → ¬ 𝜓)
Assertion
Ref Expression
nexd (𝜑 → ¬ ∃𝑥𝜓)

Proof of Theorem nexd
StepHypRef Expression
1 nexd.1 . . 3 (𝜑 → ∀𝑥𝜑)
2 nexd.2 . . 3 (𝜑 → ¬ 𝜓)
31, 2alrimih 1428 . 2 (𝜑 → ∀𝑥 ¬ 𝜓)
4 alnex 1458 . 2 (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓)
53, 4sylib 121 1 (𝜑 → ¬ ∃𝑥𝜓)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wal 1312  wex 1451
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 586  ax-in2 587  ax-5 1406  ax-gen 1408  ax-ie2 1453
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-fal 1320
This theorem is referenced by:  nexdv  1886
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