Theorem pm3.2ni 725

Description: Infer negated disjunction of negated premises. (Contributed by NM, 4-Apr-1995.)

Hypotheses

Ref	Expression
pm3.2ni.1	⊢ ¬ φ
pm3.2ni.2	⊢ ¬ ψ

Assertion

Ref	Expression
pm3.2ni	⊢ ¬ (φ ∨ ψ)

Proof of Theorem pm3.2ni

Step	Hyp	Ref	Expression
1		pm3.2ni.1	. 2 ⊢ ¬ φ
2		id 19	. . 3 ⊢ (φ → φ)
3		pm3.2ni.2	. . . 4 ⊢ ¬ ψ
4	3	pm2.21i 574	. . 3 ⊢ (ψ → φ)
5	2, 4	jaoi 635	. 2 ⊢ ((φ ∨ ψ) → φ)
6	1, 5	mto 587	1 ⊢ ¬ (φ ∨ ψ)

Syntax hints:  ¬ wn 3   ∨ wo 628

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  xrltnr  8451  pnfnlt  8458  nltmnf  8459