theorem extracted_formal_statement_13.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsFiniteMeasure μ] {X : Ω → ℝ} {t a b : ℝ} (hX : AEMeasurable X μ) (h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b)
  (ht : t < 0) (h_original_goal : Integrable (fun ω => rexp (t * X ω)) μ) :
  Integrable (fun ω => rexp (-t * -X ω)) μ := sorry

State before:
case inl
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : Ω → ℝ
t a b : ℝ
hX : AEMeasurable X μ
h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
ht : t < 0
⊢ Integrable (fun ω => rexp (t * X ω)) μ

Tactic: rw [(by ext ω; to_theorem rw [(by ring : -t * (-X ω) = t * X ω)] :
    (fun ω ↦ rexp (t * X ω)) = (fun ω ↦ rexp (-t * (-X ω))))]

State after:
[case inl
 Ω : Type u_1
 m : MeasurableSpace Ω
 μ : Measure Ω
 inst✝ : IsFiniteMeasure μ
 X : Ω → ℝ
 t a b : ℝ
 hX : AEMeasurable X μ
 h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
 ht : t < 0
 ⊢ Integrable (fun ω => rexp (-t * -X ω)) μ]

Relevant lemmas: