MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lukshef-ax1 Structured version   Visualization version   GIF version

Theorem lukshef-ax1 1696
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 1673 for its constructions.

Here, the axiom is proved as a substitution instance of nic-ax 1675. (Contributed by Anthony Hart, 31-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
lukshef-ax1 ((𝜑 ⊼ (𝜒𝜓)) ⊼ ((𝜃 ⊼ (𝜃𝜃)) ⊼ ((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃)))))

Proof of Theorem lukshef-ax1
StepHypRef Expression
1 nic-ax 1675 1 ((𝜑 ⊼ (𝜒𝜓)) ⊼ ((𝜃 ⊼ (𝜃𝜃)) ⊼ ((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃)))))
Colors of variables: wff setvar class
Syntax hints:  wnan 1482
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 210  df-an 400  df-nan 1483
This theorem is referenced by:  lukshefth1  1697  lukshefth2  1698  renicax  1699
  Copyright terms: Public domain W3C validator