Ax. 1. {P(φ)∧◻∀x[φ(x)→ψ(x)]} →P(ψ)Ax. 2.P(¬φ)↔¬P(φ)Th. 1.P(φ)→◊∃x[φ(x)]Df. 1.G(x)⟺∀φ[P(φ)→φ(x)]Ax. 3.P(G)Th. 2.◊∃xG(x)Df. 2.φ ess x⟺φ(x)∧∀ψ{ψ(x)→◻∀y[φ(y)→ψ(y)]}Ax. 4.P(φ)→◻P(φ)Th. 3.G(x)→G ess xDf. 3.E(x)⟺∀φ[φ ess x→◻∃yφ(y)]Ax. 5.P(E)Th. 4.◻∃xG(x)”.
You get it, right?
Of course.
But the caveat you intentionally did not point out (first rule of fake news) is that for the proof to be proven is that you have to accept five questionable Axioms as true to begin with.
