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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 1482. 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 1482 | . . 3 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 ⊼ (𝜓 ⊼ 𝜓))) | |
2 | 1 | bicomi 225 | . 2 ⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜓)) ↔ (𝜑 → 𝜓)) |
3 | nanbi 1484 | . 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 231 | 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 207 ⊼ wnan 1475 |
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 208 df-an 397 df-or 842 df-nan 1476 |
This theorem is referenced by: nic-stdmp 1682 nic-luk1 1683 nic-luk2 1684 nic-luk3 1685 |
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