Theorem djueq2 9901

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 2733	. 2 ⊢ 𝐶 = 𝐶
2		djueq12 9899	. 2 ⊢ ((𝐶 = 𝐶 ∧ 𝐴 = 𝐵) → (𝐶 ⊔ 𝐴) = (𝐶 ⊔ 𝐵))
3	1, 2	mpan 689	1 ⊢ (𝐴 = 𝐵 → (𝐶 ⊔ 𝐴) = (𝐶 ⊔ 𝐵))

Syntax hints:   → wi 4   = wceq 1542   ⊔ cdju 9893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-un 3954  df-opab 5212  df-xp 5683  df-dju 9896

This theorem is referenced by:  nnadju  10192