Theorem djueq1

Description: Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022)

	Ref	Expression
Assertion	djueq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ⊔ 𝐶 ) = ( 𝐵 ⊔ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ 𝐶 = 𝐶
2		djueq12	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐶 ) → ( 𝐴 ⊔ 𝐶 ) = ( 𝐵 ⊔ 𝐶 ) )
3	1 2	mpan2	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ⊔ 𝐶 ) = ( 𝐵 ⊔ 𝐶 ) )