Theorem lukshef-ax1 1619

Description: This alternative axiom for propositional calculus using the Sheffer Stroke was offered 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 1596 for its constructions.

Here, the axiom is proved as a substitution instance of nic-ax 1598. (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 1598	1 ⊢ ((𝜑 ⊼ (𝜒 ⊼ 𝜓)) ⊼ ((𝜃 ⊼ (𝜃 ⊼ 𝜃)) ⊼ ((𝜃 ⊼ 𝜒) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃)))))

Syntax hints:   ⊼ wnan 1447

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 197  df-an 386  df-nan 1448

This theorem is referenced by:  lukshefth1  1620  lukshefth2  1621  renicax  1622