Theorem uneq2

Description: Equality theorem for the union of two classes. (Contributed by NM, 5-Aug-1993)

	Ref	Expression
Assertion	uneq2	⊢ ( 𝐴 = 𝐵 → ( 𝐶 ∪ 𝐴 ) = ( 𝐶 ∪ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		uneq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ∪ 𝐶 ) = ( 𝐵 ∪ 𝐶 ) )
2		uncom	⊢ ( 𝐶 ∪ 𝐴 ) = ( 𝐴 ∪ 𝐶 )
3		uncom	⊢ ( 𝐶 ∪ 𝐵 ) = ( 𝐵 ∪ 𝐶 )
4	1 2 3	3eqtr4g	⊢ ( 𝐴 = 𝐵 → ( 𝐶 ∪ 𝐴 ) = ( 𝐶 ∪ 𝐵 ) )