Theorem djueq12 7143

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 4691	. . . 4 |- ( A = B -> ( { (/) } X. A ) = ( { (/) } X. B ) )
2	1	adantr 276	. . 3 |- ( ( A = B /\ C = D ) -> ( { (/) } X. A ) = ( { (/) } X. B ) )
3		xpeq2 4691	. . . 4 |- ( C = D -> ( { 1o } X. C ) = ( { 1o } X. D ) )
4	3	adantl 277	. . 3 |- ( ( A = B /\ C = D ) -> ( { 1o } X. C ) = ( { 1o } X. D ) )
5	2, 4	uneq12d 3328	. 2 |- ( ( A = B /\ C = D ) -> ( ( { (/) } X. A ) u. ( { 1o } X. C ) ) = ( ( { (/) } X. B ) u. ( { 1o } X. D ) ) )
6		df-dju 7142	. 2 |- ( A ⊔ C ) = ( ( { (/) } X. A ) u. ( { 1o } X. C ) )
7		df-dju 7142	. 2 |- ( B ⊔ D ) = ( ( { (/) } X. B ) u. ( { 1o } X. D ) )
8	5, 6, 7	3eqtr4g 2263	1 |- ( ( A = B /\ C = D ) -> ( A ⊔ C ) = ( B ⊔ D ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   u. cun 3164   (/)c0 3460   {csn 3633   X. cxp 4674   1oc1o 6497   ⊔ 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:  djueq1  7144  djueq2  7145  casef  7192