Theorem pm3.2ni 894

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 23	. . 3 ⊢ (𝜑 → 𝜑)
3		pm3.2ni.2	. . . 4 ⊢ ¬ 𝜓
4	3	pm2.21i 120	. . 3 ⊢ (𝜓 → 𝜑)
5	2, 4	jaoi 871	. 2 ⊢ ((𝜑 ∨ 𝜓) → 𝜑)
6	1, 5	mto 200	1 ⊢ ¬ (𝜑 ∨ 𝜓)

This proof depends on syntax axioms:  ¬ wn 3   ∨ wo 861

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-or 862

This theorem is used by:  3pm3.2ni  1519  snsn0non  6488  canthp1lem2  10731  recgt0ii  12216  xrltnr  13241  pnfnlt  13250  nltmnf  13251  smndex1n0mnd  19104  lhop  26329  2lgslem4  27726  nosgnn0  28008  axlowdimlem13  29525  clsk1indlem4  45029  clsk1indlem1  45030  dandysum2p2e4  48037