Metamath Proof Explorer

Theorem impsingle-step4

Description: Derivation of impsingle-step4 from ax-mp and impsingle . It is used as a lemma in proofs of imim1 and peirce from impsingle . It is Step 4 in Lukasiewicz, where it appears as 'CCCpqpCsp' using parenthesis-free prefix notation. (Contributed by Larry Lesyna and Jeffrey P. Machado, 2-Aug-2023) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion impsingle-step4 
|- ( ( ( ph -> ps ) -> ph ) -> ( ch -> ph ) )

Proof

Step Hyp Ref Expression
1 impsingle 
 |-  ( ( ( ta -> et ) -> ze ) -> ( ( ze -> ta ) -> ( si -> ta ) ) )
2 impsingle 
 |-  ( ( ( ph -> th ) -> ( ph -> ps ) ) -> ( ( ( ph -> ps ) -> ph ) -> ( ch -> ph ) ) )
3 impsingle 
 |-  ( ( ( ph -> ps ) -> ( ph -> ps ) ) -> ( ( ( ph -> ps ) -> ph ) -> ( ch -> ph ) ) )
4 impsingle 
 |-  ( ( ( ( ph -> ps ) -> ( ph -> ps ) ) -> ( ( ( ph -> ps ) -> ph ) -> ( ch -> ph ) ) ) -> ( ( ( ( ( ph -> ps ) -> ph ) -> ( ch -> ph ) ) -> ( ph -> ps ) ) -> ( ( ph -> th ) -> ( ph -> ps ) ) ) )
5 3 4 ax-mp 
 |-  ( ( ( ( ( ph -> ps ) -> ph ) -> ( ch -> ph ) ) -> ( ph -> ps ) ) -> ( ( ph -> th ) -> ( ph -> ps ) ) )
6 impsingle 
 |-  ( ( ( ( ( ( ph -> ps ) -> ph ) -> ( ch -> ph ) ) -> ( ph -> ps ) ) -> ( ( ph -> th ) -> ( ph -> ps ) ) ) -> ( ( ( ( ph -> th ) -> ( ph -> ps ) ) -> ( ( ( ph -> ps ) -> ph ) -> ( ch -> ph ) ) ) -> ( ( ( ( ta -> et ) -> ze ) -> ( ( ze -> ta ) -> ( si -> ta ) ) ) -> ( ( ( ph -> ps ) -> ph ) -> ( ch -> ph ) ) ) ) )
7 5 6 ax-mp 
 |-  ( ( ( ( ph -> th ) -> ( ph -> ps ) ) -> ( ( ( ph -> ps ) -> ph ) -> ( ch -> ph ) ) ) -> ( ( ( ( ta -> et ) -> ze ) -> ( ( ze -> ta ) -> ( si -> ta ) ) ) -> ( ( ( ph -> ps ) -> ph ) -> ( ch -> ph ) ) ) )
8 2 7 ax-mp 
 |-  ( ( ( ( ta -> et ) -> ze ) -> ( ( ze -> ta ) -> ( si -> ta ) ) ) -> ( ( ( ph -> ps ) -> ph ) -> ( ch -> ph ) ) )
9 1 8 ax-mp 
 |-  ( ( ( ph -> ps ) -> ph ) -> ( ch -> ph ) )