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Metamath Proof Explorer

Theorem nexdh

Description: Deduction for generalization rule for negated wff. (Contributed by NM, 2-Jan-2002)

Ref Expression
Hypotheses nexdh.1 ⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
nexdh.2 ⊢ ( 𝜑 → ¬ 𝜓 )
Assertion nexdh ( 𝜑 → ¬ ∃ 𝑥 𝜓 )

Proof

Step Hyp Ref Expression
1 nexdh.1 ⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
2 nexdh.2 ⊢ ( 𝜑 → ¬ 𝜓 )
3 1 2 alrimih ⊢ ( 𝜑 → ∀ 𝑥 ¬ 𝜓 )
4 alnex ⊢ ( ∀ 𝑥 ¬ 𝜓 ↔ ¬ ∃ 𝑥 𝜓 )
5 3 4 sylib ⊢ ( 𝜑 → ¬ ∃ 𝑥 𝜓 )