Theorem pm3.2ni 813

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 646	. . 3 ⊢ (𝜓 → 𝜑)
5	2, 4	jaoi 716	. 2 ⊢ ((𝜑 ∨ 𝜓) → 𝜑)
6	1, 5	mto 662	1 ⊢ ¬ (𝜑 ∨ 𝜓)

Syntax hints:  ¬ wn 3   ∨ wo 708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  xrltnr  9782  pnfnlt  9790  nltmnf  9791