Prove that

  ϕ ≡ ψ iff ϕ ⇔ ψ

Proof:

  The above statement can be expanded into a conjunction of two sub-statements

    ((ϕ ≡ ψ) ⇒ (ϕ ⇔ ψ)) ∧
    ((ϕ ⇔ ψ) ⇒ (ϕ ≡ ψ))

  which can be further expanded into

    ((ϕ ≡ ψ) ⇒ ((ϕ ⇒ ψ) ∧ (ψ ⇒ ϕ))) ∧
    (((ϕ ⇒ ψ) ∧ (ψ ⇒ ϕ)) ⇒ (ϕ ≡ ψ))

  1) Given: ϕ ≡ ψ
     To be proven: (ϕ ⇒ ψ) ∧ (ψ ⇒ ϕ)

     Suppose ϕ and ψ
     It follows (ϕ ⇒ ψ) ∧ (ψ ⇒ ϕ)

  2) Given: (ϕ ⇒ ψ) ∧ (ψ ⇒ ϕ)
     To be proven: ϕ ≡ ψ

     Suppose ϕ, then ψ
     Suppose ¬ϕ, then ¬ψ

     Thus ϕ ≡ ψ

Thus ϕ ≡ ψ iff ϕ ⇔ ψ

# posted by rot13(Unafba Pune) @ 8:27 AM