Theorem uneq12i

Description: Equality inference for the union of two classes. (Contributed by NM, 12-Aug-2004) (Proof shortened by Eric Schmidt, 26-Jan-2007)

	Ref	Expression
Hypotheses	uneq1i.1	⊢ 𝐴 = 𝐵
	uneq12i.2	⊢ 𝐶 = 𝐷
Assertion	uneq12i	⊢ ( 𝐴 ∪ 𝐶 ) = ( 𝐵 ∪ 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		uneq1i.1	⊢ 𝐴 = 𝐵
2		uneq12i.2	⊢ 𝐶 = 𝐷
3		uneq12	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ( 𝐴 ∪ 𝐶 ) = ( 𝐵 ∪ 𝐷 ) )
4	1 2 3	mp2an	⊢ ( 𝐴 ∪ 𝐶 ) = ( 𝐵 ∪ 𝐷 )