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Proof step 12 from proof del-addeq (previous step)

∀ a0. a0 ≠ b ∧ a0 ∈ s ↔ a0 ∈ s0, a0 = a ∨ a0 ∈ s ⊦ a = b, a0 = a, a0 = b, a ∈ s0, a ∈ s, a0 ∈ s0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 13:
a0 ∈ s, a1 ∈ s, ∀ a0. a0 ≠ b ∧ a0 ∈ s ↔ a0 ∈ s0 ⊦ a = b, a0 = b, a ∈ s0, a ∈ s, a0 ∈ s0

Rule argument

Additional information

Used simplifier rules: ⊦ ¬ a ∈ ∅; (s)% ⊦ s1 = s2 ↔ (∀ a. a ∈ s1 ↔ a ∈ s2); (s)%ne-01: ⊦ a ∈ s → s ≠ ∅; (forward) from specification set-basic