Metamath Proof Explorer

Theorem impsingle-step15

Description: Derivation of impsingle-step15 from ax-mp and impsingle . It is used as a lemma in proofs of imim1 and peirce from impsingle . It is Step 15 in Lukasiewicz, where it appears as 'CCCrqCspCCrpCsp' 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-step15 
|- ( ( ( ph -> ps ) -> ( ch -> th ) ) -> ( ( ph -> th ) -> ( ch -> th ) ) )

Proof

Step Hyp Ref Expression
1 impsingle 
 |-  ( ( ( th -> la ) -> ph ) -> ( ( ph -> th ) -> ( ch -> th ) ) )
2 impsingle 
 |-  ( ( ( ta -> si ) -> rh ) -> ( ( rh -> ta ) -> ( mu -> ta ) ) )
3 impsingle 
 |-  ( ( ( ( ( ph -> th ) -> ( ch -> th ) ) -> et ) -> ( ( th -> la ) -> ph ) ) -> ( ( ( ( th -> la ) -> ph ) -> ( ( ph -> th ) -> ( ch -> th ) ) ) -> ( ( ( ph -> ps ) -> ( ch -> th ) ) -> ( ( ph -> th ) -> ( ch -> th ) ) ) ) )
4 impsingle 
 |-  ( ( ( ( ch -> th ) -> ze ) -> ( ph -> ps ) ) -> ( ( ( ph -> ps ) -> ( ch -> th ) ) -> ( ( ph -> th ) -> ( ch -> th ) ) ) )
5 impsingle-step8 
 |-  ( ( ( ( ( ch -> th ) -> ze ) -> ( ph -> ps ) ) -> ( ( ( ph -> ps ) -> ( ch -> th ) ) -> ( ( ph -> th ) -> ( ch -> th ) ) ) ) -> ( ( ph -> ps ) -> ( ( ( ph -> ps ) -> ( ch -> th ) ) -> ( ( ph -> th ) -> ( ch -> th ) ) ) ) )
6 4 5 ax-mp 
 |-  ( ( ph -> ps ) -> ( ( ( ph -> ps ) -> ( ch -> th ) ) -> ( ( ph -> th ) -> ( ch -> th ) ) ) )
7 impsingle 
 |-  ( ( ( ph -> ps ) -> ( ( ( ph -> ps ) -> ( ch -> th ) ) -> ( ( ph -> th ) -> ( ch -> th ) ) ) ) -> ( ( ( ( ( ph -> ps ) -> ( ch -> th ) ) -> ( ( ph -> th ) -> ( ch -> th ) ) ) -> ph ) -> ( ( th -> la ) -> ph ) ) )
8 6 7 ax-mp 
 |-  ( ( ( ( ( ph -> ps ) -> ( ch -> th ) ) -> ( ( ph -> th ) -> ( ch -> th ) ) ) -> ph ) -> ( ( th -> la ) -> ph ) )
9 impsingle 
 |-  ( ( ( ( ( ( ph -> ps ) -> ( ch -> th ) ) -> ( ( ph -> th ) -> ( ch -> th ) ) ) -> ph ) -> ( ( th -> la ) -> ph ) ) -> ( ( ( ( th -> la ) -> ph ) -> ( ( ( ph -> ps ) -> ( ch -> th ) ) -> ( ( ph -> th ) -> ( ch -> th ) ) ) ) -> ( ( ( ( th -> la ) -> ph ) -> ( ( ph -> th ) -> ( ch -> th ) ) ) -> ( ( ( ph -> ps ) -> ( ch -> th ) ) -> ( ( ph -> th ) -> ( ch -> th ) ) ) ) ) )
10 8 9 ax-mp 
 |-  ( ( ( ( th -> la ) -> ph ) -> ( ( ( ph -> ps ) -> ( ch -> th ) ) -> ( ( ph -> th ) -> ( ch -> th ) ) ) ) -> ( ( ( ( th -> la ) -> ph ) -> ( ( ph -> th ) -> ( ch -> th ) ) ) -> ( ( ( ph -> ps ) -> ( ch -> th ) ) -> ( ( ph -> th ) -> ( ch -> th ) ) ) ) )
11 impsingle 
 |-  ( ( ( ( ( th -> la ) -> ph ) -> ( ( ( ph -> ps ) -> ( ch -> th ) ) -> ( ( ph -> th ) -> ( ch -> th ) ) ) ) -> ( ( ( ( th -> la ) -> ph ) -> ( ( ph -> th ) -> ( ch -> th ) ) ) -> ( ( ( ph -> ps ) -> ( ch -> th ) ) -> ( ( ph -> th ) -> ( ch -> th ) ) ) ) ) -> ( ( ( ( ( ( th -> la ) -> ph ) -> ( ( ph -> th ) -> ( ch -> th ) ) ) -> ( ( ( ph -> ps ) -> ( ch -> th ) ) -> ( ( ph -> th ) -> ( ch -> th ) ) ) ) -> ( ( th -> la ) -> ph ) ) -> ( ( ( ( ph -> th ) -> ( ch -> th ) ) -> et ) -> ( ( th -> la ) -> ph ) ) ) )
12 10 11 ax-mp 
 |-  ( ( ( ( ( ( th -> la ) -> ph ) -> ( ( ph -> th ) -> ( ch -> th ) ) ) -> ( ( ( ph -> ps ) -> ( ch -> th ) ) -> ( ( ph -> th ) -> ( ch -> th ) ) ) ) -> ( ( th -> la ) -> ph ) ) -> ( ( ( ( ph -> th ) -> ( ch -> th ) ) -> et ) -> ( ( th -> la ) -> ph ) ) )
13 impsingle 
 |-  ( ( ( ( ( ( ( th -> la ) -> ph ) -> ( ( ph -> th ) -> ( ch -> th ) ) ) -> ( ( ( ph -> ps ) -> ( ch -> th ) ) -> ( ( ph -> th ) -> ( ch -> th ) ) ) ) -> ( ( th -> la ) -> ph ) ) -> ( ( ( ( ph -> th ) -> ( ch -> th ) ) -> et ) -> ( ( th -> la ) -> ph ) ) ) -> ( ( ( ( ( ( ph -> th ) -> ( ch -> th ) ) -> et ) -> ( ( th -> la ) -> ph ) ) -> ( ( ( ( th -> la ) -> ph ) -> ( ( ph -> th ) -> ( ch -> th ) ) ) -> ( ( ( ph -> ps ) -> ( ch -> th ) ) -> ( ( ph -> th ) -> ( ch -> th ) ) ) ) ) -> ( ( ( ( ta -> si ) -> rh ) -> ( ( rh -> ta ) -> ( mu -> ta ) ) ) -> ( ( ( ( th -> la ) -> ph ) -> ( ( ph -> th ) -> ( ch -> th ) ) ) -> ( ( ( ph -> ps ) -> ( ch -> th ) ) -> ( ( ph -> th ) -> ( ch -> th ) ) ) ) ) ) )
14 12 13 ax-mp 
 |-  ( ( ( ( ( ( ph -> th ) -> ( ch -> th ) ) -> et ) -> ( ( th -> la ) -> ph ) ) -> ( ( ( ( th -> la ) -> ph ) -> ( ( ph -> th ) -> ( ch -> th ) ) ) -> ( ( ( ph -> ps ) -> ( ch -> th ) ) -> ( ( ph -> th ) -> ( ch -> th ) ) ) ) ) -> ( ( ( ( ta -> si ) -> rh ) -> ( ( rh -> ta ) -> ( mu -> ta ) ) ) -> ( ( ( ( th -> la ) -> ph ) -> ( ( ph -> th ) -> ( ch -> th ) ) ) -> ( ( ( ph -> ps ) -> ( ch -> th ) ) -> ( ( ph -> th ) -> ( ch -> th ) ) ) ) ) )
15 3 14 ax-mp 
 |-  ( ( ( ( ta -> si ) -> rh ) -> ( ( rh -> ta ) -> ( mu -> ta ) ) ) -> ( ( ( ( th -> la ) -> ph ) -> ( ( ph -> th ) -> ( ch -> th ) ) ) -> ( ( ( ph -> ps ) -> ( ch -> th ) ) -> ( ( ph -> th ) -> ( ch -> th ) ) ) ) )
16 2 15 ax-mp 
 |-  ( ( ( ( th -> la ) -> ph ) -> ( ( ph -> th ) -> ( ch -> th ) ) ) -> ( ( ( ph -> ps ) -> ( ch -> th ) ) -> ( ( ph -> th ) -> ( ch -> th ) ) ) )
17 1 16 ax-mp 
 |-  ( ( ( ph -> ps ) -> ( ch -> th ) ) -> ( ( ph -> th ) -> ( ch -> th ) ) )