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Theorem renicax 1769
Description: A rederivation of nic-ax 1745 from lukshef-ax1 1766, proving that lukshef-ax1 1766 with nic-mp 1743 can be used as a complete axiomatization of propositional calculus. (Contributed by Anthony Hart, 31-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
renicax ((𝜑 ⊼ (𝜒𝜓)) ⊼ ((𝜏 ⊼ (𝜏𝜏)) ⊼ ((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃)))))

Proof of Theorem renicax
StepHypRef Expression
1 lukshefth1 1767 . . . 4 ((((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))) ⊼ (𝜏 ⊼ (𝜏𝜏))) ⊼ (𝜑 ⊼ (𝜒𝜓)))
2 lukshefth2 1768 . . . 4 (((((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))) ⊼ (𝜏 ⊼ (𝜏𝜏))) ⊼ (𝜑 ⊼ (𝜒𝜓))) ⊼ (((𝜑 ⊼ (𝜒𝜓)) ⊼ (((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))) ⊼ (𝜏 ⊼ (𝜏𝜏)))) ⊼ ((𝜑 ⊼ (𝜒𝜓)) ⊼ (((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))) ⊼ (𝜏 ⊼ (𝜏𝜏))))))
31, 2nic-mp 1743 . . 3 ((𝜑 ⊼ (𝜒𝜓)) ⊼ (((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))) ⊼ (𝜏 ⊼ (𝜏𝜏))))
4 lukshefth2 1768 . . . 4 (((𝜏 ⊼ (𝜏𝜏)) ⊼ ((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃)))) ⊼ ((((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))) ⊼ (𝜏 ⊼ (𝜏𝜏))) ⊼ (((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))) ⊼ (𝜏 ⊼ (𝜏𝜏)))))
5 lukshef-ax1 1766 . . . 4 ((((𝜏 ⊼ (𝜏𝜏)) ⊼ ((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃)))) ⊼ ((((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))) ⊼ (𝜏 ⊼ (𝜏𝜏))) ⊼ (((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))) ⊼ (𝜏 ⊼ (𝜏𝜏))))) ⊼ (((𝜑 ⊼ (𝜒𝜓)) ⊼ ((𝜑 ⊼ (𝜒𝜓)) ⊼ (𝜑 ⊼ (𝜒𝜓)))) ⊼ (((𝜑 ⊼ (𝜒𝜓)) ⊼ (((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))) ⊼ (𝜏 ⊼ (𝜏𝜏)))) ⊼ ((((𝜏 ⊼ (𝜏𝜏)) ⊼ ((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃)))) ⊼ (𝜑 ⊼ (𝜒𝜓))) ⊼ (((𝜏 ⊼ (𝜏𝜏)) ⊼ ((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃)))) ⊼ (𝜑 ⊼ (𝜒𝜓)))))))
64, 5nic-mp 1743 . . 3 (((𝜑 ⊼ (𝜒𝜓)) ⊼ (((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))) ⊼ (𝜏 ⊼ (𝜏𝜏)))) ⊼ ((((𝜏 ⊼ (𝜏𝜏)) ⊼ ((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃)))) ⊼ (𝜑 ⊼ (𝜒𝜓))) ⊼ (((𝜏 ⊼ (𝜏𝜏)) ⊼ ((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃)))) ⊼ (𝜑 ⊼ (𝜒𝜓)))))
73, 6nic-mp 1743 . 2 (((𝜏 ⊼ (𝜏𝜏)) ⊼ ((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃)))) ⊼ (𝜑 ⊼ (𝜒𝜓)))
8 lukshefth2 1768 . 2 ((((𝜏 ⊼ (𝜏𝜏)) ⊼ ((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃)))) ⊼ (𝜑 ⊼ (𝜒𝜓))) ⊼ (((𝜑 ⊼ (𝜒𝜓)) ⊼ ((𝜏 ⊼ (𝜏𝜏)) ⊼ ((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))))) ⊼ ((𝜑 ⊼ (𝜒𝜓)) ⊼ ((𝜏 ⊼ (𝜏𝜏)) ⊼ ((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃)))))))
97, 8nic-mp 1743 1 ((𝜑 ⊼ (𝜒𝜓)) ⊼ ((𝜏 ⊼ (𝜏𝜏)) ⊼ ((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃)))))
Colors of variables: wff setvar class
Syntax hints:  wnan 1594
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 383  df-nan 1595
This theorem is referenced by: (None)
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