Theorem djueq1 7168

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 2207	. 2 ⊢ 𝐶 = 𝐶
2		djueq12 7167	. 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 1373   ⊔ cdju 7165

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-opab 4122  df-xp 4699  df-dju 7166

This theorem is referenced by:  enumct  7243  ctssexmid  7278  ctiunctal  12927  unct  12928  subctctexmid  16139  sbthom  16167