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Mirrors > Home > MPE Home > Th. List > Mathboxes > naim1 | Structured version Visualization version GIF version |
Description: Constructor theorem for ⊼. (Contributed by Anthony Hart, 1-Sep-2011.) |
Ref | Expression |
---|---|
naim1 | ⊢ ((𝜑 → 𝜓) → ((𝜓 ⊼ 𝜒) → (𝜑 ⊼ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | con3 156 | . . 3 ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑)) | |
2 | 1 | orim1d 965 | . 2 ⊢ ((𝜑 → 𝜓) → ((¬ 𝜓 ∨ ¬ 𝜒) → (¬ 𝜑 ∨ ¬ 𝜒))) |
3 | pm3.13 994 | . . . 4 ⊢ (¬ (𝜓 ∧ 𝜒) → (¬ 𝜓 ∨ ¬ 𝜒)) | |
4 | pm3.14 995 | . . . 4 ⊢ ((¬ 𝜑 ∨ ¬ 𝜒) → ¬ (𝜑 ∧ 𝜒)) | |
5 | 3, 4 | imim12i 62 | . . 3 ⊢ (((¬ 𝜓 ∨ ¬ 𝜒) → (¬ 𝜑 ∨ ¬ 𝜒)) → (¬ (𝜓 ∧ 𝜒) → ¬ (𝜑 ∧ 𝜒))) |
6 | df-nan 1487 | . . 3 ⊢ ((𝜓 ⊼ 𝜒) ↔ ¬ (𝜓 ∧ 𝜒)) | |
7 | df-nan 1487 | . . 3 ⊢ ((𝜑 ⊼ 𝜒) ↔ ¬ (𝜑 ∧ 𝜒)) | |
8 | 5, 6, 7 | 3imtr4g 299 | . 2 ⊢ (((¬ 𝜓 ∨ ¬ 𝜒) → (¬ 𝜑 ∨ ¬ 𝜒)) → ((𝜓 ⊼ 𝜒) → (𝜑 ⊼ 𝜒))) |
9 | 2, 8 | syl 17 | 1 ⊢ ((𝜑 → 𝜓) → ((𝜓 ⊼ 𝜒) → (𝜑 ⊼ 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∨ wo 846 ⊼ wnan 1486 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-nan 1487 |
This theorem is referenced by: naim1i 34236 |
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