simp[LogicalConnective.bex, eq_ex_iff]
L : Language
L₁ : Language
L₂ : Language
L₃ : Language
ξ : Type u_1
ξ₁ : Type u_2
ξ₂ : Type u_3
ξ₃ : Type u_4
n n₁ n₂✝ n₂ m m₁ m₂ m₃ : ℕ
ω : Rew L ξ₁ n₁ ξ₂ n₂
p : Semiformula L ξ₁ n₁
q₁ q₂ : Semiformula L ξ₂ (n₂ + 1)
⊢ (ω.hom p = ∃[q₁] q₂) ↔ ∃ p₁ p₂, ω.q.hom p₁ = q₁ ∧ ω.q.hom p₂ = q₂ ∧ p = ∃[p₁] p₂
L : Language
L₁ : Language
L₂ : Language
L₃ : Language
ξ : Type u_1
ξ₁ : Type u_2
ξ₂ : Type u_3
ξ₃ : Type u_4
n n₁ n₂✝ n₂ m m₁ m₂ m₃ : ℕ
ω : Rew L ξ₁ n₁ ξ₂ n₂
p : Semiformula L ξ₁ n₁
q₁ q₂ : Semiformula L ξ₂ (n₂ + 1)
⊢ (∃ p', (∃ p₁, ω.q.hom p₁ = q₁ ∧ ∃ x, ω.q.hom x = q₂ ∧ p' = p₁ ⋏ x) ∧ p = ∃' p') ↔
    ∃ p₁, ω.q.hom p₁ = q₁ ∧ ∃ x, ω.q.hom x = q₂ ∧ p = ∃' p₁ ⋏ x