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Ax.1. P(φ)∧☐∀[φ(x)→ψ(x)]→P(ψ)
Ax.2. P(¬φ)↔¬P(φ)
Th.1. P(φ)→◇∃[φ(x)]
Df.1. G(x)⇔∀φ[P(φ)→φ(x)]
Ax.3. P(G)
Th.2. ◇∃x G(x)
Df.2. φ ess x⇔φ(x)∧∀ψ{ψ(x)→☐∀x[φ(x)→ψ(x)]}
Ax.4. P(φ)→☐P(φ)
Th.3. G(x)→G ess x
Df.3. E(x)⇔∀φ[φ ess x→☐∃x φ(x)]
Ax.5. P(E)
Th.4. ☐∃x G(x)