Theorem djueq1 7238

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

Assertion

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

Proof of Theorem djueq1

Step	Hyp	Ref	Expression
1		eqid 2231	. 2 ⊢ 𝐶 = 𝐶
2		djueq12 7237	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐶) → (𝐴 ⊔ 𝐶) = (𝐵 ⊔ 𝐶))
3	1, 2	mpan2 425	1 ⊢ (𝐴 = 𝐵 → (𝐴 ⊔ 𝐶) = (𝐵 ⊔ 𝐶))

Syntax hints:   → wi 4   = wceq 1397   ⊔ cdju 7235

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-opab 4151  df-xp 4731  df-dju 7236

This theorem is referenced by:  enumct  7313  ctssexmid  7348  ctiunctal  13061  unct  13062  subctctexmid  16601  sbthom  16630