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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 569 mto 588 pm5.19 622 noel 3228 0nelop 3985 elirr 4266 en2lp 4278 soirri 4719 0neqopab 5550 fzp1disj 8942 fzonel 9016 fzouzdisj 9036 |
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