Theorem pm3.2ni 899

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 116	. . 3 ⊢ (𝜓 → 𝜑)
5	2, 4	jaoi 394	. 2 ⊢ ((𝜑 ∨ 𝜓) → 𝜑)
6	1, 5	mto 188	1 ⊢ ¬ (𝜑 ∨ 𝜓)

Syntax hints:  ¬ wn 3   ∨ wo 383

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 197  df-or 385

This theorem is referenced by:  snsn0non  5846  canthp1lem2  9475  recgt0ii  10929  xrltnr  11953  pnfnlt  11962  nltmnf  11963  lhop  23779  2lgslem4  25131  axlowdimlem13  25834  3pm3.2ni  31594  nosgnn0  31811  clsk1indlem4  38342  clsk1indlem1  38343  dandysum2p2e4  41165