Theorem djueq1 7099

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 2193	. 2 |- C = C
2		djueq12 7098	. 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 1364   ⊔ cdju 7096

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-opab 4091  df-xp 4665  df-dju 7097

This theorem is referenced by:  enumct  7174  ctssexmid  7209  ctiunctal  12598  unct  12599  subctctexmid  15491  sbthom  15516