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Metamath Proof Explorer

Theorem elintd

Description: Membership in class intersection. (Contributed by Glauco Siliprandi, 3-Jan-2021)

Ref Expression
Hypotheses elintd.1 ⊢ Ⅎ x φ
elintd.2 ⊢ φ → A ∈ V
elintd.3 ⊢ φ ∧ x ∈ B → A ∈ x
Assertion elintd ⊢ φ → A ∈ ⋂ B

Proof

Step Hyp Ref Expression
1 elintd.1 ⊢ Ⅎ x φ
2 elintd.2 ⊢ φ → A ∈ V
3 elintd.3 ⊢ φ ∧ x ∈ B → A ∈ x
4 3 ex ⊢ φ → x ∈ B → A ∈ x
5 1 4 ralrimi ⊢ φ → ∀ x ∈ B A ∈ x
6 elintg ⊢ A ∈ V → A ∈ ⋂ B ↔ ∀ x ∈ B A ∈ x
7 2 6 syl ⊢ φ → A ∈ ⋂ B ↔ ∀ x ∈ B A ∈ x
8 5 7 mpbird ⊢ φ → A ∈ ⋂ B