Theorem nexdh 1589

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 1565	. 2 ⊢ (φ → ∀x ¬ ψ)
4		alnex 1543	. 2 ⊢ (∀x ¬ ψ ↔ ¬ ∃xψ)
5	3, 4	sylib 188	1 ⊢ (φ → ¬ ∃xψ)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540  ∃wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177  df-ex 1542

This theorem is referenced by:  nexd  1771