... ((( A 1 ) → A 7) → (A 8→ A 1 )), then S(ϕ)includes A 1 , A 7, A 8,( A 1 ), (A 8→ A 1 ), (( A 1 ) → A 7), and ((( A 1 ) → A 7) → (A 8→ A 1 )) itself.Note that if you write out a formula ... A 1 ,A 4, ( A 1 ), (A 1 ∨ A 4), (A 4→ (A 1 ∨ A 4)) } .(As an exercise, where did ( A 1 )comefrom?) Problem 1. 11. Find all the subformulas of each of the followingformulas. (1) (¬(( A 56 ) ... β)).Replacing α, β,andγ by particular formulas of LP in any one of theschemas A1 , A2 , or A3 gives an axiom of LP.For example, (A 1 → (A 4→ A 1 )) is an axiom, being an instance ofaxiom schema...