Theorem nexd 780

Description: Deduction for generalization rule for negated wff.

Hypotheses

Ref	Expression
nexd.1	⊢ (φ → ∀xφ)
nexd.2	⊢ (φ → ¬ ψ)

Assertion

Ref	Expression
nexd	⊢ (φ → ¬ ∃xψ)

Proof of Theorem nexd

Step	Hyp	Ref	Expression
1		nexd.1	. . 3 ⊢ (φ → ∀xφ)
2		nexd.2	. . 3 ⊢ (φ → ¬ ψ)
3	1, 2	19.21ai 740	. 2 ⊢ (φ → ∀x ¬ ψ)
4		alnex 716	. 2 ⊢ (∀x ¬ ψ ↔ ¬ ∃xψ)
5	3, 4	sylib 173	1 ⊢ (φ → ¬ ∃xψ)

Syntax hints:  ¬ wn 1   → wi 2  ∀wal 672  ∃wex 678

This theorem is referenced by:  nexdv 983  axrepnd 3740  axunndlem1 3741  axunnd 3742

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-ex 679

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