|   | 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 119 | . . 3 ⊢ (𝜓 → 𝜑) | 
| 5 | 2, 4 | jaoi 858 | . 2 ⊢ ((𝜑 ∨ 𝜓) → 𝜑) | 
| 6 | 1, 5 | mto 197 | 1 ⊢ ¬ (𝜑 ∨ 𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ∨ wo 848 | 
| 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 207 df-or 849 | 
| This theorem is referenced by: 3pm3.2ni 1490 snsn0non 6509 canthp1lem2 10693 recgt0ii 12174 xrltnr 13161 pnfnlt 13170 nltmnf 13171 smndex1n0mnd 18925 lhop 26055 2lgslem4 27450 nosgnn0 27703 axlowdimlem13 28969 clsk1indlem4 44057 clsk1indlem1 44058 dandysum2p2e4 47010 | 
| Copyright terms: Public domain | W3C validator |