Examples of Substitution

Example 1. Write down the proof in L for the following well-formed formula:

((A → (B → C)) → (A → B)) → ((A → (B → C)) → (A → C))

Proof. We shall construct proof in L.

We begin with Axiom 2: 

(A → (B → C)) → ((A → B) → (A → C)) ___(1)

Now substitute as follows:

                                        (A → (B → C))     for     A, 

                                        (A → B)                for     B, 

                                        (A → C)                for     C. 

And we get:

((A → (B → C)) → ((A → B) → (A → C))) → (((A → (B → C)) → (A → B)) → ((A → (B → C))→ (A → C))) ___(2)

Now apply the law of detachment to expression (1) and expression (2), we get:

((A → (B → C)) → (A → B)) → ((A → (B → C)) → (A → C))

This is what to be shown.

And this completes the proof.

Example 2. Write down the proof in L for the following well-formed formula:

(A → (B → ((A → B)))

We begin with Axiom 1:

A → (B → A) ___(1)

Now substitute as follows:

                                         (B → (A → B))     for     A 

                              and                           A     for     B.

And we get:

(B → (A → B))→ (A → (B → (A → B))) ___(2)

Now apply the law of detachment to (1) and (2), we get:

(A → (B → (A → B)))

This is what to be shown.

And this completes the proof