Metamath Proof Explorer

Theorem elunant

Description: A statement is true for every element of the union of a pair of classes if and only if it is true for every element of the first class and for every element of the second class. (Contributed by BTernaryTau, 27-Sep-2023)

Ref Expression
Assertion elunant ⊢ C ∈ A ∪ B → φ ↔ C ∈ A → φ ∧ C ∈ B → φ

Proof

Step Hyp Ref Expression
1 elun ⊢ C ∈ A ∪ B ↔ C ∈ A ∨ C ∈ B
2 1 imbi1i ⊢ C ∈ A ∪ B → φ ↔ C ∈ A ∨ C ∈ B → φ
3 jaob ⊢ C ∈ A ∨ C ∈ B → φ ↔ C ∈ A → φ ∧ C ∈ B → φ
4 2 3 bitri ⊢ C ∈ A ∪ B → φ ↔ C ∈ A → φ ∧ C ∈ B → φ