New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > nic-dfim | GIF version |
Description: Define implication in terms of 'nand'. Analogous to ((φ ⊼ (ψ ⊼ ψ)) ↔ (φ → ψ)). 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 1292 | . . 3 ⊢ ((φ → ψ) ↔ (φ ⊼ (ψ ⊼ ψ))) | |
2 | 1 | bicomi 193 | . 2 ⊢ ((φ ⊼ (ψ ⊼ ψ)) ↔ (φ → ψ)) |
3 | nanbi 1294 | . 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 199 | 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 176 ⊼ wnan 1287 |
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 177 df-or 359 df-an 360 df-nan 1288 |
This theorem is referenced by: nic-stdmp 1455 nic-luk1 1456 nic-luk2 1457 nic-luk3 1458 |
Copyright terms: Public domain | W3C validator |