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Metamath Proof Explorer

Theorem raleqbi1dv

Description: Equality deduction for restricted universal quantifier. (Contributed by NM, 16-Nov-1995) (Proof shortened by Steven Nguyen, 5-May-2023)

Ref Expression
Hypothesis raleqbi1dv.1 ⊢ A = B → φ ↔ ψ
Assertion raleqbi1dv ⊢ A = B → ∀ x ∈ A φ ↔ ∀ x ∈ B ψ

Proof

Step Hyp Ref Expression
1 raleqbi1dv.1 ⊢ A = B → φ ↔ ψ
2 id ⊢ A = B → A = B
3 2 1 raleqbidvv ⊢ A = B → ∀ x ∈ A φ ↔ ∀ x ∈ B ψ