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 e. ( A u. B ) -> ph ) <-> ( ( C e. A -> ph ) /\ ( C e. B -> ph ) ) )

Proof

Step	Hyp	Ref	Expression
1		elun	|- ( C e. ( A u. B ) <-> ( C e. A \/ C e. B ) )
2	1	imbi1i	|- ( ( C e. ( A u. B ) -> ph ) <-> ( ( C e. A \/ C e. B ) -> ph ) )
3		jaob	|- ( ( ( C e. A \/ C e. B ) -> ph ) <-> ( ( C e. A -> ph ) /\ ( C e. B -> ph ) ) )
4	2 3	bitri	|- ( ( C e. ( A u. B ) -> ph ) <-> ( ( C e. A -> ph ) /\ ( C e. B -> ph ) ) )