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Proof step 6 from proof subset-app-03 (previous step)

⊦ (∀ a. a ∈ y → a ∈ x ∨ a ∈ z) ↔ (∀ a. a ∈ y → a ∈ x + z)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 7:
a ∈ y, ∀ a. a ∈ y → a ∈ x ∨ a ∈ z ⊦ a ∈ x, a ∈ z

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list%in-append: ⊦ a ∈ x + y ↔ ¬ ¬(a ∈ x ∨ a ∈ y); from specification list