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Metamath Proof Explorer

Theorem ralbidv

Description: Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 20-Nov-1994) Reduce dependencies on axioms. (Revised by Wolf Lammen, 5-Dec-2019)

Ref Expression
Hypothesis ralbidv.1 ⊢ φ → ψ ↔ χ
Assertion ralbidv ⊢ φ → ∀ x ∈ A ψ ↔ ∀ x ∈ A χ

Proof

Step Hyp Ref Expression
1 ralbidv.1 ⊢ φ → ψ ↔ χ
2 1 adantr ⊢ φ ∧ x ∈ A → ψ ↔ χ
3 2 ralbidva ⊢ φ → ∀ x ∈ A ψ ↔ ∀ x ∈ A χ