Theorem pm3.2ni 880

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 22	. . 3 ⊢ (𝜑 → 𝜑)
3		pm3.2ni.2	. . . 4 ⊢ ¬ 𝜓
4	3	pm2.21i 119	. . 3 ⊢ (𝜓 → 𝜑)
5	2, 4	jaoi 856	. 2 ⊢ ((𝜑 ∨ 𝜓) → 𝜑)
6	1, 5	mto 196	1 ⊢ ¬ (𝜑 ∨ 𝜓)

Syntax hints:  ¬ wn 3   ∨ wo 846

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

This theorem depends on definitions:  df-bi 206  df-or 847

This theorem is referenced by:  3pm3.2ni  1489  snsn0non  6490  canthp1lem2  10648  recgt0ii  12120  xrltnr  13099  pnfnlt  13108  nltmnf  13109  smndex1n0mnd  18793  lhop  25533  2lgslem4  26909  nosgnn0  27161  axlowdimlem13  28212  clsk1indlem4  42795  clsk1indlem1  42796  dandysum2p2e4  45708