Metamath Proof Explorer

Theorem impsingle-step18

Description: Derivation of impsingle-step18 from ax-mp and impsingle . It is used as a lemma in proofs of imim1 and peirce from impsingle . It is Step 18 in Lukasiewicz, where it appears as 'CCCCrpCspCCCpqrtCuCCCpqrt' 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-step18 
|- ( ( ( ( ph -> ps ) -> ( ch -> ps ) ) -> ( ( ( ps -> th ) -> ph ) -> ta ) ) -> ( et -> ( ( ( ps -> th ) -> ph ) -> ta ) ) )

Proof

Step Hyp Ref Expression
1 impsingle 
 |-  ( ( ( ps -> th ) -> ph ) -> ( ( ph -> ps ) -> ( ch -> ps ) ) )
2 impsingle 
 |-  ( ( ( ( ch -> ps ) -> rh ) -> ( ( ( ps -> th ) -> ph ) -> ta ) ) -> ( ( ( ( ( ps -> th ) -> ph ) -> ta ) -> ( ch -> ps ) ) -> ( ( ph -> ps ) -> ( ch -> ps ) ) ) )
3 impsingle-step8 
 |-  ( ( ( ( ( ch -> ps ) -> rh ) -> ( ( ( ps -> th ) -> ph ) -> ta ) ) -> ( ( ( ( ( ps -> th ) -> ph ) -> ta ) -> ( ch -> ps ) ) -> ( ( ph -> ps ) -> ( ch -> ps ) ) ) ) -> ( ( ( ( ps -> th ) -> ph ) -> ta ) -> ( ( ( ( ( ps -> th ) -> ph ) -> ta ) -> ( ch -> ps ) ) -> ( ( ph -> ps ) -> ( ch -> ps ) ) ) ) )
4 2 3 ax-mp 
 |-  ( ( ( ( ps -> th ) -> ph ) -> ta ) -> ( ( ( ( ( ps -> th ) -> ph ) -> ta ) -> ( ch -> ps ) ) -> ( ( ph -> ps ) -> ( ch -> ps ) ) ) )
5 impsingle-step15 
 |-  ( ( ( ( ( ps -> th ) -> ph ) -> ta ) -> ( ( ( ( ( ps -> th ) -> ph ) -> ta ) -> ( ch -> ps ) ) -> ( ( ph -> ps ) -> ( ch -> ps ) ) ) ) -> ( ( ( ( ps -> th ) -> ph ) -> ( ( ph -> ps ) -> ( ch -> ps ) ) ) -> ( ( ( ( ( ps -> th ) -> ph ) -> ta ) -> ( ch -> ps ) ) -> ( ( ph -> ps ) -> ( ch -> ps ) ) ) ) )
6 4 5 ax-mp 
 |-  ( ( ( ( ps -> th ) -> ph ) -> ( ( ph -> ps ) -> ( ch -> ps ) ) ) -> ( ( ( ( ( ps -> th ) -> ph ) -> ta ) -> ( ch -> ps ) ) -> ( ( ph -> ps ) -> ( ch -> ps ) ) ) )
7 1 6 ax-mp 
 |-  ( ( ( ( ( ps -> th ) -> ph ) -> ta ) -> ( ch -> ps ) ) -> ( ( ph -> ps ) -> ( ch -> ps ) ) )
8 impsingle 
 |-  ( ( ( ( ( ( ps -> th ) -> ph ) -> ta ) -> ( ch -> ps ) ) -> ( ( ph -> ps ) -> ( ch -> ps ) ) ) -> ( ( ( ( ph -> ps ) -> ( ch -> ps ) ) -> ( ( ( ps -> th ) -> ph ) -> ta ) ) -> ( et -> ( ( ( ps -> th ) -> ph ) -> ta ) ) ) )
9 7 8 ax-mp 
 |-  ( ( ( ( ph -> ps ) -> ( ch -> ps ) ) -> ( ( ( ps -> th ) -> ph ) -> ta ) ) -> ( et -> ( ( ( ps -> th ) -> ph ) -> ta ) ) )