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Metamath Proof Explorer

Theorem 2ralbidva

Description: Formula-building rule for restricted universal quantifiers (deduction form). (Contributed by NM, 4-Mar-1997) Reduce dependencies on axioms. (Revised by Wolf Lammen, 9-Dec-2019)

Ref Expression
Hypothesis 2ralbidva.1 ⊢ φ ∧ x ∈ A ∧ y ∈ B → ψ ↔ χ
Assertion 2ralbidva ⊢ φ → ∀ x ∈ A ∀ y ∈ B ψ ↔ ∀ x ∈ A ∀ y ∈ B χ

Proof

Step Hyp Ref Expression
1 2ralbidva.1 ⊢ φ ∧ x ∈ A ∧ y ∈ B → ψ ↔ χ
2 1 anassrs ⊢ φ ∧ x ∈ A ∧ y ∈ B → ψ ↔ χ
3 2 ralbidva ⊢ φ ∧ x ∈ A → ∀ y ∈ B ψ ↔ ∀ y ∈ B χ
4 3 ralbidva ⊢ φ → ∀ x ∈ A ∀ y ∈ B ψ ↔ ∀ x ∈ A ∀ y ∈ B χ