Theorem un4

Description: A rearrangement of the union of 4 classes. (Contributed by NM, 12-Aug-2004)

	Ref	Expression
Assertion	un4	$$⊢\left(A\cup B\right)\cup \left(C\cup D\right)=\left(A\cup C\right)\cup \left(B\cup D\right)$$

Proof

Step	Hyp	Ref	Expression
1		un12	$$⊢B\cup \left(C\cup D\right)=C\cup \left(B\cup D\right)$$
2	1	uneq2i	$$⊢A\cup \left(B\cup \left(C\cup D\right)\right)=A\cup \left(C\cup \left(B\cup D\right)\right)$$
3		unass	$$⊢\left(A\cup B\right)\cup \left(C\cup D\right)=A\cup \left(B\cup \left(C\cup D\right)\right)$$
4		unass	$$⊢\left(A\cup C\right)\cup \left(B\cup D\right)=A\cup \left(C\cup \left(B\cup D\right)\right)$$
5	2 3 4	3eqtr4i	$$⊢\left(A\cup B\right)\cup \left(C\cup D\right)=\left(A\cup C\right)\cup \left(B\cup D\right)$$