# Monotone formulas: a necessary and sufficient condition in propositional logic

I want to show that a wff φ\varphi is monotne iff or all atomic pp and for every wff α\alpha and β\beta satisfying α⊨β\alpha \models \beta we have ϕ[α/p]⊨ϕ[β/p]\phi[\alpha/p] \models \phi[\beta/p].

I succeeded in proving the “only if” part of this statement by induction. But I have problem in proving the “if” part. Could you please provide me with the idea for this part?

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