![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nic-dfim | Structured version Visualization version GIF version |
Description: This theorem "defines" implication in terms of 'nand'. Analogous to nanim 1566. In a pure (standalone) treatment of Nicod's axiom, this theorem would be changed to a definition ($a statement). (Contributed by NM, 11-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nic-dfim | ⊢ (((𝜑 ⊼ (𝜓 ⊼ 𝜓)) ⊼ (𝜑 → 𝜓)) ⊼ (((𝜑 ⊼ (𝜓 ⊼ 𝜓)) ⊼ (𝜑 ⊼ (𝜓 ⊼ 𝜓))) ⊼ ((𝜑 → 𝜓) ⊼ (𝜑 → 𝜓)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nanim 1566 | . . 3 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 ⊼ (𝜓 ⊼ 𝜓))) | |
2 | 1 | bicomi 216 | . 2 ⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜓)) ↔ (𝜑 → 𝜓)) |
3 | nanbi 1569 | . 2 ⊢ (((𝜑 ⊼ (𝜓 ⊼ 𝜓)) ↔ (𝜑 → 𝜓)) ↔ (((𝜑 ⊼ (𝜓 ⊼ 𝜓)) ⊼ (𝜑 → 𝜓)) ⊼ (((𝜑 ⊼ (𝜓 ⊼ 𝜓)) ⊼ (𝜑 ⊼ (𝜓 ⊼ 𝜓))) ⊼ ((𝜑 → 𝜓) ⊼ (𝜑 → 𝜓))))) | |
4 | 2, 3 | mpbi 222 | 1 ⊢ (((𝜑 ⊼ (𝜓 ⊼ 𝜓)) ⊼ (𝜑 → 𝜓)) ⊼ (((𝜑 ⊼ (𝜓 ⊼ 𝜓)) ⊼ (𝜑 ⊼ (𝜓 ⊼ 𝜓))) ⊼ ((𝜑 → 𝜓) ⊼ (𝜑 → 𝜓)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ⊼ wnan 1557 |
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 199 df-an 387 df-or 837 df-nan 1558 |
This theorem is referenced by: nic-stdmp 1734 nic-luk1 1735 nic-luk2 1736 nic-luk3 1737 |
Copyright terms: Public domain | W3C validator |