![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > pm2.65i | GIF version |
Description: Inference rule for proof by contradiction. (Contributed by NM, 18-May-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.) |
Ref | Expression |
---|---|
pm2.65i.1 | ⊢ (φ → ψ) |
pm2.65i.2 | ⊢ (φ → ¬ ψ) |
Ref | Expression |
---|---|
pm2.65i | ⊢ ¬ φ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.65i.2 | . . 3 ⊢ (φ → ¬ ψ) | |
2 | pm2.65i.1 | . . 3 ⊢ (φ → ψ) | |
3 | 1, 2 | nsyl3 556 | . 2 ⊢ (φ → ¬ φ) |
4 | pm2.01 546 | . 2 ⊢ ((φ → ¬ φ) → ¬ φ) | |
5 | 3, 4 | ax-mp 7 | 1 ⊢ ¬ φ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-in1 544 ax-in2 545 |
This theorem is referenced by: mt2 568 mto 587 pm5.19 621 noel 3222 0nelop 3976 elirr 4224 en2lp 4232 soirri 4662 0neqopab 5492 fzp1disj 8712 fzonel 8786 fzouzdisj 8806 |
Copyright terms: Public domain | W3C validator |