Mathbox for Anthony Hart |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > naim2 | Structured version Visualization version GIF version |
Description: Constructor theorem for ⊼. (Contributed by Anthony Hart, 1-Sep-2011.) |
Ref | Expression |
---|---|
naim2 | ⊢ ((𝜑 → 𝜓) → ((𝜒 ⊼ 𝜓) → (𝜒 ⊼ 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | con3 149 | . . 3 ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑)) | |
2 | 1 | orim2d 885 | . 2 ⊢ ((𝜑 → 𝜓) → ((¬ 𝜒 ∨ ¬ 𝜓) → (¬ 𝜒 ∨ ¬ 𝜑))) |
3 | pm3.13 522 | . . . 4 ⊢ (¬ (𝜒 ∧ 𝜓) → (¬ 𝜒 ∨ ¬ 𝜓)) | |
4 | pm3.14 523 | . . . 4 ⊢ ((¬ 𝜒 ∨ ¬ 𝜑) → ¬ (𝜒 ∧ 𝜑)) | |
5 | 3, 4 | imim12i 62 | . . 3 ⊢ (((¬ 𝜒 ∨ ¬ 𝜓) → (¬ 𝜒 ∨ ¬ 𝜑)) → (¬ (𝜒 ∧ 𝜓) → ¬ (𝜒 ∧ 𝜑))) |
6 | df-nan 1448 | . . 3 ⊢ ((𝜒 ⊼ 𝜓) ↔ ¬ (𝜒 ∧ 𝜓)) | |
7 | df-nan 1448 | . . 3 ⊢ ((𝜒 ⊼ 𝜑) ↔ ¬ (𝜒 ∧ 𝜑)) | |
8 | 5, 6, 7 | 3imtr4g 285 | . 2 ⊢ (((¬ 𝜒 ∨ ¬ 𝜓) → (¬ 𝜒 ∨ ¬ 𝜑)) → ((𝜒 ⊼ 𝜓) → (𝜒 ⊼ 𝜑))) |
9 | 2, 8 | syl 17 | 1 ⊢ ((𝜑 → 𝜓) → ((𝜒 ⊼ 𝜓) → (𝜒 ⊼ 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 383 ∧ wa 384 ⊼ wnan 1447 |
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 197 df-or 385 df-an 386 df-nan 1448 |
This theorem is referenced by: naim2i 32387 |
Copyright terms: Public domain | W3C validator |