theorem extracted_formal_statement_0.{u_4, u_3, u_2, u_1} {E : Type u_1} [inst : NormedAddCommGroup E]
  [inst_1 : NormedSpace ℝ E] {Ω : Type u_2} {β : Type u_3} {β' : Type u_4} {_mΩ : MeasurableSpace Ω} {μ : Measure Ω}
  {f : Ω → β} {g : Ω → β'} {F : β × β' → E} {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'}
  [inst_2 : IsFiniteMeasure μ] {hF' : StronglyMeasurable F} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
  (hfg : IndepFun f g μ) (hF : HasFiniteIntegral F ((Measure.map f μ).prod (Measure.map g μ))) :
  let ν := Measure.map f μ;
  let ν' := Measure.map g μ;
  Measure.map (fun ω => (f ω, g ω)) μ = ν.prod ν' →
    ∫ (ω : Ω), ∫ (x' : β'), F (f ω, x') ∂ν' ∂μ = ∫ (x : β), ∫ (x' : β'), F (x, x') ∂ν' ∂ν := sorry

State before:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
Ω : Type u_2
β : Type u_3
β' : Type u_4
_mΩ : MeasurableSpace Ω
μ : Measure Ω
f : Ω → β
g : Ω → β'
F : β × β' → E
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
inst✝ : IsFiniteMeasure μ
hF' : StronglyMeasurable F
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hfg : IndepFun f g μ
hF : HasFiniteIntegral F ((Measure.map f μ).prod (Measure.map g μ))
ν : Measure β := Measure.map f μ
ν' : Measure β' := Measure.map g μ
hfg' : Measure.map (fun ω => (f ω, g ω)) μ = ν.prod ν'
⊢ ∫ (ω : Ω), ∫ (x' : β'), F (f ω, x') ∂ν' ∂μ = ∫ (x : β), ∫ (x' : β'), F (x, x') ∂ν' ∂ν

Tactic: apply Eq.symm
    apply MeasureTheory.integral_map
    assumption
    apply StronglyMeasurable.aestronglyMeasurable
    exact StronglyMeasurable.integral_prod_right hF'

State after:

Relevant lemmas:
MeasureTheory.integral_map : ∀ {α : Type u_1} {G : Type u_5} [inst : NormedAddCommGroup G] [inst_3 : NormedSpace ℝ G]
  {m : MeasurableSpace α} {μ : Measure α} {β : Type u_7} [inst_4 : MeasurableSpace β] {φ : α → β},
  AEMeasurable φ μ →
    ∀ {f : β → G}, AEStronglyMeasurable f (Measure.map φ μ) → ∫ (y : β), f y ∂Measure.map φ μ = ∫ (x : α), f (φ x) ∂μ
Eq.symm : ∀ {α : Sort u} {a b : α}, a = b → b = a
MeasureTheory.StronglyMeasurable.aestronglyMeasurable : ∀ {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace β]
  {m m₀ : MeasurableSpace α} {μ : Measure α} {f : α → β}, StronglyMeasurable f → AEStronglyMeasurable f μ
MeasureTheory.StronglyMeasurable.integral_prod_right : ∀ {α : Type u_1} {β : Type u_2} {E : Type u_3}
  [inst : MeasurableSpace α] [inst_3 : MeasurableSpace β] {ν : Measure β} [inst_4 : NormedAddCommGroup E]
  [inst_5 : NormedSpace ℝ E] [inst_6 : SFinite ν] ⦃f : α → β → E⦄,
  StronglyMeasurable (Function.uncurry f) → StronglyMeasurable fun x => ∫ (y : β), f x y ∂ν