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Mirrors > Home > ILE Home > Th. List > pm2.65i | GIF version |
Description: Inference 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 616 | . 2 ⊢ (𝜑 → ¬ 𝜑) |
4 | pm2.01 606 | . 2 ⊢ ((𝜑 → ¬ 𝜑) → ¬ 𝜑) | |
5 | 3, 4 | ax-mp 5 | 1 ⊢ ¬ 𝜑 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-in1 604 ax-in2 605 |
This theorem is referenced by: mt2 630 mto 652 pm5.19 696 noel 3411 0nelop 4223 elirr 4515 en2lp 4528 soirri 4995 canth 5793 0neqopab 5881 fzp1disj 10009 fzonel 10089 fzouzdisj 10109 bj-imnimnn 13513 nnnotnotr 13765 |
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