Theorem uneq12

Description: Equality theorem for the union of two classes. (Contributed by NM, 29-Mar-1998)

	Ref	Expression
Assertion	uneq12	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ( 𝐴 ∪ 𝐶 ) = ( 𝐵 ∪ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		uneq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ∪ 𝐶 ) = ( 𝐵 ∪ 𝐶 ) )
2		uneq2	⊢ ( 𝐶 = 𝐷 → ( 𝐵 ∪ 𝐶 ) = ( 𝐵 ∪ 𝐷 ) )
3	1 2	sylan9eq	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ( 𝐴 ∪ 𝐶 ) = ( 𝐵 ∪ 𝐷 ) )