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 ) ) )