theorem extracted_formal_statement_209 {φ : ℝ → ℝ} {η x y : ℝ} (hx : x ∈ Ioo 0 η) (hy : y ∈ Ioo 0 η)
  (h1 : ConcaveOn ℝ (Ico 0 η) φ) (left : φ 0 = 0) (left_1 : ContDiffOn ℝ 1 φ (Ioo 0 η)) (left_2 : ContinuousAt φ 0)
  (h6 : ∀ x ∈ Ioo 0 η, deriv φ x > 0) (hdiff : DifferentiableAt ℝ φ x) :
  let hx' := Ioo_subset_Ico_self hx;
  let hy' := Ioo_subset_Ico_self hy;
  x = y → φ y - φ y ≥ deriv φ y * (y - y) := sorry

State before:
case intro.intro.intro.intro.inl
φ : ℝ → ℝ
η x y : ℝ
hx : x ∈ Ioo 0 η
hy : y ∈ Ioo 0 η
h1 : ConcaveOn ℝ (Ico 0 η) φ
left✝² : φ 0 = 0
left✝¹ : ContDiffOn ℝ 1 φ (Ioo 0 η)
left✝ : ContinuousAt φ 0
h6 : ∀ x ∈ Ioo 0 η, deriv φ x > 0
hdiff : DifferentiableAt ℝ φ x
hx' : x ∈ Ico 0 η := Ioo_subset_Ico_self hx
hy' : y ∈ Ico 0 η := Ioo_subset_Ico_self hy
heq : x = y
⊢ φ y - φ y ≥ deriv φ y * (y - y)

Tactic: simp only [sub_self, mul_zero, ge_iff_le, le_refl]

State after:

Relevant lemmas:
le_refl : ∀ {α : Type u_1} [inst : Preorder α] (a : α), a ≤ a
sub_self : ∀ {G : Type u_1} [inst : AddGroup G] (a : G), a - a = 0
ge_iff_le : ∀ {α : Type u} [inst : LE α] {x y : α}, x ≥ y ↔ y ≤ x
MulZeroClass.mul_zero : ∀ {M₀ : Type u} [self : MulZeroClass M₀] (a : M₀), a * 0 = 0