Theorem djueq2 9866

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

Assertion

Ref	Expression
djueq2	⊢ (𝐴 = 𝐵 → (𝐶 ⊔ 𝐴) = (𝐶 ⊔ 𝐵))

Proof of Theorem djueq2

Step	Hyp	Ref	Expression
1		eqid 2764	. 2 ⊢ 𝐶 = 𝐶
2		djueq12 9864	. 2 ⊢ ((𝐶 = 𝐶 ∧ 𝐴 = 𝐵) → (𝐶 ⊔ 𝐴) = (𝐶 ⊔ 𝐵))
3	1, 2	mpan 700	1 ⊢ (𝐴 = 𝐵 → (𝐶 ⊔ 𝐴) = (𝐶 ⊔ 𝐵))

Syntax hints:   → wi 4   = wceq 1562   ⊔ cdju 9858

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1565  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-v 3458  df-un 3911  df-opab 5165  df-xp 5655  df-dju 9861

This theorem is referenced by:  nnadju  10156