Theorem nexdh 1872

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

Step	Hyp	Ref	Expression
1		nexdh.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
2		nexdh.2	. . 3 ⊢ (𝜑 → ¬ 𝜓)
3	1, 2	alrimih 1830	. 2 ⊢ (𝜑 → ∀𝑥 ¬ 𝜓)
4		alnex 1788	. 2 ⊢ (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓)
5	3, 4	sylib 221	1 ⊢ (𝜑 → ¬ ∃𝑥𝜓)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540  ∃wex 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816

This theorem depends on definitions:  df-bi 210  df-ex 1787

This theorem is referenced by:  nexdv  1943  nexd  2223