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Proof step 7 from proof diff-unchanged (previous step)

∀ y. x0 \ y = x0 ↔ (∀ a. ¬ a ∈ x0 ∨ ¬ a ∈ y) ⊦ a ∈ y, (a ' + x0) \ y = a ' + x0 ↔ (∀ a0. a0 ≠ a ∧ ¬ a0 ∈ x0 ∨ ¬ a0 ∈ y)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 8:
∀ y. x0 \ y = x0 ↔ (∀ a. ¬ a ∈ x0 ∨ ¬ a ∈ y) ⊦ x0 \ y = x0 ↔ (∀ a0. ¬ a0 ∈ x0 ∨ ¬ a0 ∈ y), a ∈ y

Rule argument

Additional information

Used simplifier rules: diff-n: ⊦ ¬ a ∈ y → (a ' + x) \ y = a + (x \ y);%app-08: ⊦ a ' + x = a ' + y ↔ x = y; from specification list%a: ⊦ (x + y) + z = x + y + z; from specification list%c: ⊦ a + x = a ' + x; from specification list