Theorem djueq12 7098

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 4674	. . . 4 |- ( A = B -> ( { (/) } X. A ) = ( { (/) } X. B ) )
2	1	adantr 276	. . 3 |- ( ( A = B /\ C = D ) -> ( { (/) } X. A ) = ( { (/) } X. B ) )
3		xpeq2 4674	. . . 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 3314	. 2 |- ( ( A = B /\ C = D ) -> ( ( { (/) } X. A ) u. ( { 1o } X. C ) ) = ( ( { (/) } X. B ) u. ( { 1o } X. D ) ) )
6		df-dju 7097	. 2 |- ( A ⊔ C ) = ( ( { (/) } X. A ) u. ( { 1o } X. C ) )
7		df-dju 7097	. 2 |- ( B ⊔ D ) = ( ( { (/) } X. B ) u. ( { 1o } X. D ) )
8	5, 6, 7	3eqtr4g 2251	1 |- ( ( A = B /\ C = D ) -> ( A ⊔ C ) = ( B ⊔ D ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   u. cun 3151   (/)c0 3446   {csn 3618   X. cxp 4657   1oc1o 6462   ⊔ 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:  djueq1  7099  djueq2  7100  casef  7147