Theorem djueq1 7144

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 2205	. 2 |- C = C
2		djueq12 7143	. 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 7141

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-opab 4107  df-xp 4682  df-dju 7142

This theorem is referenced by:  enumct  7219  ctssexmid  7254  ctiunctal  12845  unct  12846  subctctexmid  15974  sbthom  16002