Description: This alternative axiom for propositional calculus using the Sheffer Stroke was discovered by Lukasiewicz in his Selected Works. It improves on Nicod's axiom by reducing its number of variables by one.
This axiom also uses nic-mp for its constructions.
Here, the axiom is proved as a substitution instance of nic-ax . (Contributed by Anthony Hart, 31-Jul-2011) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | lukshef-ax1 | ⊢ ( ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜓 ) ) ⊼ ( ( 𝜃 ⊼ ( 𝜃 ⊼ 𝜃 ) ) ⊼ ( ( 𝜃 ⊼ 𝜒 ) ⊼ ( ( 𝜑 ⊼ 𝜃 ) ⊼ ( 𝜑 ⊼ 𝜃 ) ) ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nic-ax | ⊢ ( ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜓 ) ) ⊼ ( ( 𝜃 ⊼ ( 𝜃 ⊼ 𝜃 ) ) ⊼ ( ( 𝜃 ⊼ 𝜒 ) ⊼ ( ( 𝜑 ⊼ 𝜃 ) ⊼ ( 𝜑 ⊼ 𝜃 ) ) ) ) ) |