Theorem djueq1 6996

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 2164	. 2 |- C = C
2		djueq12 6995	. 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 1342   ⊔ cdju 6993

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2723  df-un 3115  df-opab 4038  df-xp 4604  df-dju 6994

This theorem is referenced by:  enumct  7071  ctssexmid  7105  ctiunctal  12311  unct  12312  subctctexmid  13715  sbthom  13739