Metamath Proof Explorer

Theorem renicax

Description: A rederivation of nic-ax from lukshef-ax1 , proving that lukshef-ax1 with nic-mp 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.)

Ref Expression
Assertion renicax 
|- ( ( ph -/\ ( ch -/\ ps ) ) -/\ ( ( ta -/\ ( ta -/\ ta ) ) -/\ ( ( th -/\ ch ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) ) )

Proof

Step Hyp Ref Expression
1 lukshefth1 
 |-  ( ( ( ( th -/\ ch ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) -/\ ( ta -/\ ( ta -/\ ta ) ) ) -/\ ( ph -/\ ( ch -/\ ps ) ) )
2 lukshefth2 
 |-  ( ( ( ( ( th -/\ ch ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) -/\ ( ta -/\ ( ta -/\ ta ) ) ) -/\ ( ph -/\ ( ch -/\ ps ) ) ) -/\ ( ( ( ph -/\ ( ch -/\ ps ) ) -/\ ( ( ( th -/\ ch ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) -/\ ( ta -/\ ( ta -/\ ta ) ) ) ) -/\ ( ( ph -/\ ( ch -/\ ps ) ) -/\ ( ( ( th -/\ ch ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) -/\ ( ta -/\ ( ta -/\ ta ) ) ) ) ) )
3 1 2 nic-mp 
 |-  ( ( ph -/\ ( ch -/\ ps ) ) -/\ ( ( ( th -/\ ch ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) -/\ ( ta -/\ ( ta -/\ ta ) ) ) )
4 lukshefth2 
 |-  ( ( ( ta -/\ ( ta -/\ ta ) ) -/\ ( ( th -/\ ch ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) ) -/\ ( ( ( ( th -/\ ch ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) -/\ ( ta -/\ ( ta -/\ ta ) ) ) -/\ ( ( ( th -/\ ch ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) -/\ ( ta -/\ ( ta -/\ ta ) ) ) ) )
5 lukshef-ax1 
 |-  ( ( ( ( ta -/\ ( ta -/\ ta ) ) -/\ ( ( th -/\ ch ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) ) -/\ ( ( ( ( th -/\ ch ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) -/\ ( ta -/\ ( ta -/\ ta ) ) ) -/\ ( ( ( th -/\ ch ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) -/\ ( ta -/\ ( ta -/\ ta ) ) ) ) ) -/\ ( ( ( ph -/\ ( ch -/\ ps ) ) -/\ ( ( ph -/\ ( ch -/\ ps ) ) -/\ ( ph -/\ ( ch -/\ ps ) ) ) ) -/\ ( ( ( ph -/\ ( ch -/\ ps ) ) -/\ ( ( ( th -/\ ch ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) -/\ ( ta -/\ ( ta -/\ ta ) ) ) ) -/\ ( ( ( ( ta -/\ ( ta -/\ ta ) ) -/\ ( ( th -/\ ch ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) ) -/\ ( ph -/\ ( ch -/\ ps ) ) ) -/\ ( ( ( ta -/\ ( ta -/\ ta ) ) -/\ ( ( th -/\ ch ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) ) -/\ ( ph -/\ ( ch -/\ ps ) ) ) ) ) ) )
6 4 5 nic-mp 
 |-  ( ( ( ph -/\ ( ch -/\ ps ) ) -/\ ( ( ( th -/\ ch ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) -/\ ( ta -/\ ( ta -/\ ta ) ) ) ) -/\ ( ( ( ( ta -/\ ( ta -/\ ta ) ) -/\ ( ( th -/\ ch ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) ) -/\ ( ph -/\ ( ch -/\ ps ) ) ) -/\ ( ( ( ta -/\ ( ta -/\ ta ) ) -/\ ( ( th -/\ ch ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) ) -/\ ( ph -/\ ( ch -/\ ps ) ) ) ) )
7 3 6 nic-mp 
 |-  ( ( ( ta -/\ ( ta -/\ ta ) ) -/\ ( ( th -/\ ch ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) ) -/\ ( ph -/\ ( ch -/\ ps ) ) )
8 lukshefth2 
 |-  ( ( ( ( ta -/\ ( ta -/\ ta ) ) -/\ ( ( th -/\ ch ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) ) -/\ ( ph -/\ ( ch -/\ ps ) ) ) -/\ ( ( ( ph -/\ ( ch -/\ ps ) ) -/\ ( ( ta -/\ ( ta -/\ ta ) ) -/\ ( ( th -/\ ch ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) ) ) -/\ ( ( ph -/\ ( ch -/\ ps ) ) -/\ ( ( ta -/\ ( ta -/\ ta ) ) -/\ ( ( th -/\ ch ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) ) ) ) )
9 7 8 nic-mp 
 |-  ( ( ph -/\ ( ch -/\ ps ) ) -/\ ( ( ta -/\ ( ta -/\ ta ) ) -/\ ( ( th -/\ ch ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) ) )