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Mirrors > Home > MPE Home > Th. List > nic-mp | Structured version Visualization version GIF version |
Description: Derive Nicod's rule of modus ponens using 'nand', from the standard one. Although the major and minor premise together also imply 𝜒, this form is necessary for useful derivations from nic-ax 1670. In a pure (standalone) treatment of Nicod's axiom, this theorem would be changed to an axiom ($a statement). (Contributed by Jeff Hoffman, 19-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nic-jmin | ⊢ 𝜑 |
nic-jmaj | ⊢ (𝜑 ⊼ (𝜒 ⊼ 𝜓)) |
Ref | Expression |
---|---|
nic-mp | ⊢ 𝜓 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nic-jmin | . 2 ⊢ 𝜑 | |
2 | nic-jmaj | . . . 4 ⊢ (𝜑 ⊼ (𝜒 ⊼ 𝜓)) | |
3 | nannan 1486 | . . . 4 ⊢ ((𝜑 ⊼ (𝜒 ⊼ 𝜓)) ↔ (𝜑 → (𝜒 ∧ 𝜓))) | |
4 | 2, 3 | mpbi 232 | . . 3 ⊢ (𝜑 → (𝜒 ∧ 𝜓)) |
5 | 4 | simprd 498 | . 2 ⊢ (𝜑 → 𝜓) |
6 | 1, 5 | ax-mp 5 | 1 ⊢ 𝜓 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ⊼ wnan 1480 |
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 209 df-an 399 df-nan 1481 |
This theorem is referenced by: nic-imp 1672 nic-idlem2 1674 nic-id 1675 nic-swap 1676 nic-isw1 1677 nic-isw2 1678 nic-iimp1 1679 nic-idel 1681 nic-ich 1682 nic-stdmp 1687 nic-luk1 1688 nic-luk2 1689 nic-luk3 1690 lukshefth1 1692 lukshefth2 1693 renicax 1694 |
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