L : Language
T : Theory L
α : Type w
n : ℕ
φ ψ : BoundedFormula L α (n + 1)
h : SemanticallyEquivalent T φ ψ
⊢ ∀ (M : ModelType T) (v : α → ↑M) (xs : Fin n → ↑M),
    (∃ a, BoundedFormula.Realize φ v (Fin.snoc xs a)) ↔ ∃ a, BoundedFormula.Realize ψ v (Fin.snoc xs a)
L : Language
T : Theory L
α : Type w
n : ℕ
φ ψ : BoundedFormula L α (n + 1)
h : SemanticallyEquivalent T φ ψ
⊢ SemanticallyEquivalent T (∃'φ) (∃'ψ)
simp_rw [SemanticallyEquivalent, ModelsBoundedFormula, BoundedFormula.realize_iff,
  BoundedFormula.realize_ex]
no goals
L : Language
T : Theory L
α : Type w
n : ℕ
φ ψ : BoundedFormula L α (n + 1)
h : SemanticallyEquivalent T φ ψ
⊢ ∀ (M : ModelType T) (v : α → ↑M) (xs : Fin n → ↑M),
    (∃ a, BoundedFormula.Realize φ v (Fin.snoc xs a)) ↔ ∃ a, BoundedFormula.Realize ψ v (Fin.snoc xs a)
exact fun M v xs => exists_congr fun a => h.realize_bd_iff