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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 627 | . 2 ⊢ (𝜑 → ¬ 𝜑) |
4 | pm2.01 617 | . 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 615 ax-in2 616 |
This theorem is referenced by: mt2 641 mto 663 pm5.19 707 noel 3446 0nelop 4273 elirr 4565 en2lp 4578 soirri 5048 canth 5859 0neqopab 5951 fzp1disj 10132 fzonel 10213 fzouzdisj 10233 4sqlem17 12519 lgsval2lem 15054 bj-imnimnn 15154 nnnotnotr 15406 |
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