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Proof step 24 from proof compl-monotone (previous step)

f(λ x0. ¬ S(x0)) ⊆ f(λ x0. ¬ S0(x0)), f(λ x0. ¬ S(x0))(x), S0 ⊆ S ⊦ f(λ x0. ¬ S0(x0))(x)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 25:
⊦ S ⊆ S0 ↔ (∀ x. S(x) → S0(x))
Proof step 26:
∀ x. f(λ x0. ¬ S(x0))(x) → f(λ x0. ¬ S0(x0))(x), f(λ x0. ¬ S(x0))(x), S0 ⊆ S ⊦ f(λ x0. ¬ S0(x0))(x)

Rule argument

spec: inst: name: subseteq-def seq: ⊦ S ⊆ S0 ↔ (∀ x. S(x) → S0(x)) subst: [S0, S] → [f(λ x0. ¬ S0(x0)), f(λ x0. ¬ S(x0))] flags: rewrite, current

Additional information

Used terms: f(λ x0. ¬ S0(x0)), f(λ x0. ¬ S(x0))