Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > pm3.2ni | Structured version Visualization version GIF version |
Description: Infer negated disjunction of negated premises. (Contributed by NM, 4-Apr-1995.) |
Ref | Expression |
---|---|
pm3.2ni.1 | ⊢ ¬ 𝜑 |
pm3.2ni.2 | ⊢ ¬ 𝜓 |
Ref | Expression |
---|---|
pm3.2ni | ⊢ ¬ (𝜑 ∨ 𝜓) |
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 115 | . . 3 ⊢ (𝜓 → 𝜑) |
5 | 2, 4 | jaoi 393 | . 2 ⊢ ((𝜑 ∨ 𝜓) → 𝜑) |
6 | 1, 5 | mto 187 | 1 ⊢ ¬ (𝜑 ∨ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 382 |
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 196 df-or 384 |
This theorem is referenced by: snsn0non 5763 canthp1lem2 9354 recgt0ii 10808 xrltnr 11829 pnfnlt 11838 nltmnf 11839 lhop 23583 2lgslem4 24931 axlowdimlem13 25634 3pm3.2ni 30849 nosgnn0 31055 clsk1indlem4 37362 clsk1indlem1 37363 dandysum2p2e4 39814 |
Copyright terms: Public domain | W3C validator |