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 ∈ ( 𝐴𝐵 ) → 𝜑 ) ↔ ( ( 𝐶𝐴𝜑 ) ∧ ( 𝐶𝐵𝜑 ) ) )

Proof

Step Hyp Ref Expression
1 elun ( 𝐶 ∈ ( 𝐴𝐵 ) ↔ ( 𝐶𝐴𝐶𝐵 ) )
2 1 imbi1i ( ( 𝐶 ∈ ( 𝐴𝐵 ) → 𝜑 ) ↔ ( ( 𝐶𝐴𝐶𝐵 ) → 𝜑 ) )
3 jaob ( ( ( 𝐶𝐴𝐶𝐵 ) → 𝜑 ) ↔ ( ( 𝐶𝐴𝜑 ) ∧ ( 𝐶𝐵𝜑 ) ) )
4 2 3 bitri ( ( 𝐶 ∈ ( 𝐴𝐵 ) → 𝜑 ) ↔ ( ( 𝐶𝐴𝜑 ) ∧ ( 𝐶𝐵𝜑 ) ) )