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