Theorem un4

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

	Ref	Expression
Assertion	un4	⊢ ( ( 𝐴 ∪ 𝐵 ) ∪ ( 𝐶 ∪ 𝐷 ) ) = ( ( 𝐴 ∪ 𝐶 ) ∪ ( 𝐵 ∪ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		un12	⊢ ( 𝐵 ∪ ( 𝐶 ∪ 𝐷 ) ) = ( 𝐶 ∪ ( 𝐵 ∪ 𝐷 ) )
2	1	uneq2i	⊢ ( 𝐴 ∪ ( 𝐵 ∪ ( 𝐶 ∪ 𝐷 ) ) ) = ( 𝐴 ∪ ( 𝐶 ∪ ( 𝐵 ∪ 𝐷 ) ) )
3		unass	⊢ ( ( 𝐴 ∪ 𝐵 ) ∪ ( 𝐶 ∪ 𝐷 ) ) = ( 𝐴 ∪ ( 𝐵 ∪ ( 𝐶 ∪ 𝐷 ) ) )
4		unass	⊢ ( ( 𝐴 ∪ 𝐶 ) ∪ ( 𝐵 ∪ 𝐷 ) ) = ( 𝐴 ∪ ( 𝐶 ∪ ( 𝐵 ∪ 𝐷 ) ) )
5	2 3 4	3eqtr4i	⊢ ( ( 𝐴 ∪ 𝐵 ) ∪ ( 𝐶 ∪ 𝐷 ) ) = ( ( 𝐴 ∪ 𝐶 ) ∪ ( 𝐵 ∪ 𝐷 ) )