Theorem djueq2 7300

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 2231	. 2 ⊢ 𝐶 = 𝐶
2		djueq12 7298	. 2 ⊢ ((𝐶 = 𝐶 ∧ 𝐴 = 𝐵) → (𝐶 ⊔ 𝐴) = (𝐶 ⊔ 𝐵))
3	1, 2	mpan 424	1 ⊢ (𝐴 = 𝐵 → (𝐶 ⊔ 𝐴) = (𝐶 ⊔ 𝐵))

Syntax hints:   → wi 4   = wceq 1398   ⊔ cdju 7296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-opab 4156  df-xp 4737  df-dju 7297

This theorem is referenced by: (None)