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Theorem djueq12 7014
Description: Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
Assertion
Ref Expression
djueq12  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A C )  =  ( B D ) )

Proof of Theorem djueq12
StepHypRef Expression
1 xpeq2 4624 . . . 4  |-  ( A  =  B  ->  ( { (/) }  X.  A
)  =  ( {
(/) }  X.  B
) )
21adantr 274 . . 3  |-  ( ( A  =  B  /\  C  =  D )  ->  ( { (/) }  X.  A )  =  ( { (/) }  X.  B
) )
3 xpeq2 4624 . . . 4  |-  ( C  =  D  ->  ( { 1o }  X.  C
)  =  ( { 1o }  X.  D
) )
43adantl 275 . . 3  |-  ( ( A  =  B  /\  C  =  D )  ->  ( { 1o }  X.  C )  =  ( { 1o }  X.  D ) )
52, 4uneq12d 3282 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  ( ( { (/) }  X.  A )  u.  ( { 1o }  X.  C ) )  =  ( ( { (/) }  X.  B )  u.  ( { 1o }  X.  D ) ) )
6 df-dju 7013 . 2  |-  ( A C )  =  ( ( { (/) }  X.  A )  u.  ( { 1o }  X.  C
) )
7 df-dju 7013 . 2  |-  ( B D )  =  ( ( { (/) }  X.  B )  u.  ( { 1o }  X.  D
) )
85, 6, 73eqtr4g 2228 1  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A C )  =  ( B D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1348    u. cun 3119   (/)c0 3414   {csn 3581    X. cxp 4607   1oc1o 6386   ⊔ cdju 7012
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-opab 4049  df-xp 4615  df-dju 7013
This theorem is referenced by:  djueq1  7015  djueq2  7016  casef  7063
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