 Proof: Let σ(x,y,z) be a formula that numeralwise expresses the number theoretic predicate ‘y is the Gödel number of the formula obtained by replacing the variable v0 in the formula whose Gödel number is x by the term z’. Let θ(v0) be the formula ∃v1(φ(v1) ∧ σ(v0, v1, v0)). Let k = ⌈θ(v0)⌉ and ψ = θ(k). Now directly by the construction P ⊢ ψ ↔ φ(⌈ψ⌉).