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Metamath Proof Explorer

Theorem ralbida

Description: Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 6-Oct-2003) (Proof shortened by Wolf Lammen, 31-Oct-2024)

Ref Expression
Hypotheses ralbida.1 ⊢ Ⅎ x φ
ralbida.2 ⊢ φ ∧ x ∈ A → ψ ↔ χ
Assertion ralbida ⊢ φ → ∀ x ∈ A ψ ↔ ∀ x ∈ A χ

Proof

Step Hyp Ref Expression
1 ralbida.1 ⊢ Ⅎ x φ
2 ralbida.2 ⊢ φ ∧ x ∈ A → ψ ↔ χ
3 2 biimpd ⊢ φ ∧ x ∈ A → ψ → χ
4 1 3 ralimdaa ⊢ φ → ∀ x ∈ A ψ → ∀ x ∈ A χ
5 2 biimprd ⊢ φ ∧ x ∈ A → χ → ψ
6 1 5 ralimdaa ⊢ φ → ∀ x ∈ A χ → ∀ x ∈ A ψ
7 4 6 impbid ⊢ φ → ∀ x ∈ A ψ ↔ ∀ x ∈ A χ