Theorem lukshef-ax1 1695

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 1672 for its constructions.

Here, the axiom is proved as a substitution instance of nic-ax 1674. (Contributed by Anthony Hart, 31-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
lukshef-ax1	⊢ ((𝜑 ⊼ (𝜒 ⊼ 𝜓)) ⊼ ((𝜃 ⊼ (𝜃 ⊼ 𝜃)) ⊼ ((𝜃 ⊼ 𝜒) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃)))))

Proof of Theorem lukshef-ax1

Step	Hyp	Ref	Expression
1		nic-ax 1674	1 ⊢ ((𝜑 ⊼ (𝜒 ⊼ 𝜓)) ⊼ ((𝜃 ⊼ (𝜃 ⊼ 𝜃)) ⊼ ((𝜃 ⊼ 𝜒) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃)))))

Syntax hints:   ⊼ wnan 1481

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 1482

This theorem is referenced by:  lukshefth1  1696  lukshefth2  1697  renicax  1698