Theorem djueq1 7005

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 2165	. 2 |- C = C
2		djueq12 7004	. 2 |- ( ( A = B /\ C = C ) -> ( A ⊔ C ) = ( B ⊔ C ) )
3	1, 2	mpan2 422	1 |- ( A = B -> ( A ⊔ C ) = ( B ⊔ C ) )

Syntax hints:   -> wi 4   = wceq 1343   ⊔ cdju 7002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-opab 4044  df-xp 4610  df-dju 7003

This theorem is referenced by:  enumct  7080  ctssexmid  7114  ctiunctal  12374  unct  12375  subctctexmid  13881  sbthom  13905