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Theorem lukshef-ax1 1609
 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 1586 for its constructions. Here, the axiom is proved as a substitution instance of nic-ax 1588. (Contributed by Anthony Hart, 31-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
lukshef-ax1 ((𝜑 ⊼ (𝜒𝜓)) ⊼ ((𝜃 ⊼ (𝜃𝜃)) ⊼ ((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃)))))

Proof of Theorem lukshef-ax1
StepHypRef Expression
1 nic-ax 1588 1 ((𝜑 ⊼ (𝜒𝜓)) ⊼ ((𝜃 ⊼ (𝜃𝜃)) ⊼ ((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃)))))
 Colors of variables: wff setvar class Syntax hints:   ⊼ wnan 1438 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 195  df-an 384  df-nan 1439 This theorem is referenced by:  lukshefth1  1610  lukshefth2  1611  renicax  1612
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