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Mirrors > Home > ILE Home > Th. List > mtord | GIF version |
Description: A modus tollens deduction involving disjunction. (Contributed by Jeff Hankins, 15-Jul-2009.) (Revised by Mario Carneiro, 31-Jan-2015.) |
Ref | Expression |
---|---|
mtord.1 | ⊢ (𝜑 → ¬ 𝜒) |
mtord.2 | ⊢ (𝜑 → ¬ 𝜃) |
mtord.3 | ⊢ (𝜑 → (𝜓 → (𝜒 ∨ 𝜃))) |
Ref | Expression |
---|---|
mtord | ⊢ (𝜑 → ¬ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mtord.3 | . . 3 ⊢ (𝜑 → (𝜓 → (𝜒 ∨ 𝜃))) | |
2 | mtord.1 | . . . . 5 ⊢ (𝜑 → ¬ 𝜒) | |
3 | 2 | pm2.21d 614 | . . . 4 ⊢ (𝜑 → (𝜒 → ¬ 𝜓)) |
4 | mtord.2 | . . . . 5 ⊢ (𝜑 → ¬ 𝜃) | |
5 | 4 | pm2.21d 614 | . . . 4 ⊢ (𝜑 → (𝜃 → ¬ 𝜓)) |
6 | 3, 5 | jaod 712 | . . 3 ⊢ (𝜑 → ((𝜒 ∨ 𝜃) → ¬ 𝜓)) |
7 | 1, 6 | syld 45 | . 2 ⊢ (𝜑 → (𝜓 → ¬ 𝜓)) |
8 | 7 | pm2.01d 613 | 1 ⊢ (𝜑 → ¬ 𝜓) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 703 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: swoer 6538 |
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