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Mirrors > Home > MPE Home > Th. List > minimp | Structured version Visualization version GIF version |
Description: A single axiom for minimal implicational calculus, due to Meredith. Other single axioms of the same length are known, but it is thought to be the minimal length. (Contributed by BJ, 4-Apr-2021.) |
Ref | Expression |
---|---|
minimp | ⊢ (𝜑 → ((𝜓 → 𝜒) → (((𝜃 → 𝜓) → (𝜒 → 𝜏)) → (𝜓 → 𝜏)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | jarr 106 | . . . 4 ⊢ (((𝜃 → 𝜓) → (𝜒 → 𝜏)) → (𝜓 → (𝜒 → 𝜏))) | |
2 | 1 | a2d 29 | . . 3 ⊢ (((𝜃 → 𝜓) → (𝜒 → 𝜏)) → ((𝜓 → 𝜒) → (𝜓 → 𝜏))) |
3 | 2 | com12 32 | . 2 ⊢ ((𝜓 → 𝜒) → (((𝜃 → 𝜓) → (𝜒 → 𝜏)) → (𝜓 → 𝜏))) |
4 | 3 | a1i 11 | 1 ⊢ (𝜑 → ((𝜓 → 𝜒) → (((𝜃 → 𝜓) → (𝜒 → 𝜏)) → (𝜓 → 𝜏)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 |
This theorem is referenced by: minimp-sylsimp 1561 minimp-ax2c 1563 |
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