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Metamath Proof Explorer

Theorem r19.27v

Description: Restricted quantitifer version of one direction of 19.27 . (Assuming F/_ x A , the other direction holds when A is nonempty, see r19.27zv .) (Contributed by NM, 3-Jun-2004) (Proof shortened by Andrew Salmon, 30-May-2011) (Proof shortened by Wolf Lammen, 17-Jun-2023)

Ref Expression
Assertion r19.27v ⊢ ∀ x ∈ A φ ∧ ψ → ∀ x ∈ A φ ∧ ψ

Proof

Step Hyp Ref Expression
1 id ⊢ ψ → ψ
2 1 ralrimivw ⊢ ψ → ∀ x ∈ A ψ
3 2 anim2i ⊢ ∀ x ∈ A φ ∧ ψ → ∀ x ∈ A φ ∧ ∀ x ∈ A ψ
4 r19.26 ⊢ ∀ x ∈ A φ ∧ ψ ↔ ∀ x ∈ A φ ∧ ∀ x ∈ A ψ
5 3 4 sylibr ⊢ ∀ x ∈ A φ ∧ ψ → ∀ x ∈ A φ ∧ ψ