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Theorem nexdh 1789
Description: Deduction for generalization rule for negated wff. (Contributed by NM, 2-Jan-2002.)
Hypotheses
Ref Expression
nexdh.1 (𝜑 → ∀𝑥𝜑)
nexdh.2 (𝜑 → ¬ 𝜓)
Assertion
Ref Expression
nexdh (𝜑 → ¬ ∃𝑥𝜓)

Proof of Theorem nexdh
StepHypRef Expression
1 nexdh.1 . . 3 (𝜑 → ∀𝑥𝜑)
2 nexdh.2 . . 3 (𝜑 → ¬ 𝜓)
31, 2alrimih 1748 . 2 (𝜑 → ∀𝑥 ¬ 𝜓)
4 alnex 1703 . 2 (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓)
53, 4sylib 208 1 (𝜑 → ¬ ∃𝑥𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1478  wex 1701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734
This theorem depends on definitions:  df-bi 197  df-ex 1702
This theorem is referenced by:  nexdv  1861  nexd  2087  nexdOLD  2197
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