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Metamath Proof Explorer

Theorem r19.28v

Description: Restricted quantifier version of one direction of 19.28 . (Assuming F/_ x A , the other direction holds when A is nonempty, see r19.28zv .) (Contributed by NM, 2-Apr-2004) (Proof shortened by Wolf Lammen, 17-Jun-2023)

Ref Expression
Assertion r19.28v ⊢ φ ∧ ∀ x ∈ A ψ → ∀ x ∈ A φ ∧ ψ

Proof

Step Hyp Ref Expression
1 id ⊢ φ → φ
2 1 ralrimivw ⊢ φ → ∀ x ∈ A φ
3 2 anim1i ⊢ φ ∧ ∀ x ∈ A ψ → ∀ x ∈ A φ ∧ ∀ x ∈ A ψ
4 r19.26 ⊢ ∀ x ∈ A φ ∧ ψ ↔ ∀ x ∈ A φ ∧ ∀ x ∈ A ψ
5 3 4 sylibr ⊢ φ ∧ ∀ x ∈ A ψ → ∀ x ∈ A φ ∧ ψ