Theorem djueq12 7004

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

Step	Hyp	Ref	Expression
1		xpeq2 4619	. . . 4 |- ( A = B -> ( { (/) } X. A ) = ( { (/) } X. B ) )
2	1	adantr 274	. . 3 |- ( ( A = B /\ C = D ) -> ( { (/) } X. A ) = ( { (/) } X. B ) )
3		xpeq2 4619	. . . 4 |- ( C = D -> ( { 1o } X. C ) = ( { 1o } X. D ) )
4	3	adantl 275	. . 3 |- ( ( A = B /\ C = D ) -> ( { 1o } X. C ) = ( { 1o } X. D ) )
5	2, 4	uneq12d 3277	. 2 |- ( ( A = B /\ C = D ) -> ( ( { (/) } X. A ) u. ( { 1o } X. C ) ) = ( ( { (/) } X. B ) u. ( { 1o } X. D ) ) )
6		df-dju 7003	. 2 |- ( A ⊔ C ) = ( ( { (/) } X. A ) u. ( { 1o } X. C ) )
7		df-dju 7003	. 2 |- ( B ⊔ D ) = ( ( { (/) } X. B ) u. ( { 1o } X. D ) )
8	5, 6, 7	3eqtr4g 2224	1 |- ( ( A = B /\ C = D ) -> ( A ⊔ C ) = ( B ⊔ D ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   u. cun 3114   (/)c0 3409   {csn 3576   X. cxp 4602   1oc1o 6377   ⊔ 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:  djueq1  7005  djueq2  7006  casef  7053