Theorem djueq12 7237

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 4740	. . . 4 |- ( A = B -> ( { (/) } X. A ) = ( { (/) } X. B ) )
2	1	adantr 276	. . 3 |- ( ( A = B /\ C = D ) -> ( { (/) } X. A ) = ( { (/) } X. B ) )
3		xpeq2 4740	. . . 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 3362	. 2 |- ( ( A = B /\ C = D ) -> ( ( { (/) } X. A ) u. ( { 1o } X. C ) ) = ( ( { (/) } X. B ) u. ( { 1o } X. D ) ) )
6		df-dju 7236	. 2 |- ( A ⊔ C ) = ( ( { (/) } X. A ) u. ( { 1o } X. C ) )
7		df-dju 7236	. 2 |- ( B ⊔ D ) = ( ( { (/) } X. B ) u. ( { 1o } X. D ) )
8	5, 6, 7	3eqtr4g 2289	1 |- ( ( A = B /\ C = D ) -> ( A ⊔ C ) = ( B ⊔ D ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   u. cun 3198   (/)c0 3494   {csn 3669   X. cxp 4723   1oc1o 6574   ⊔ cdju 7235

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-opab 4151  df-xp 4731  df-dju 7236

This theorem is referenced by:  djueq1  7238  djueq2  7239  casef  7286