Theorem djueq12 7016

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 4626	. . . 4 |- ( A = B -> ( { (/) } X. A ) = ( { (/) } X. B ) )
2	1	adantr 274	. . 3 |- ( ( A = B /\ C = D ) -> ( { (/) } X. A ) = ( { (/) } X. B ) )
3		xpeq2 4626	. . . 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 3282	. 2 |- ( ( A = B /\ C = D ) -> ( ( { (/) } X. A ) u. ( { 1o } X. C ) ) = ( ( { (/) } X. B ) u. ( { 1o } X. D ) ) )
6		df-dju 7015	. 2 |- ( A ⊔ C ) = ( ( { (/) } X. A ) u. ( { 1o } X. C ) )
7		df-dju 7015	. 2 |- ( B ⊔ D ) = ( ( { (/) } X. B ) u. ( { 1o } X. D ) )
8	5, 6, 7	3eqtr4g 2228	1 |- ( ( A = B /\ C = D ) -> ( A ⊔ C ) = ( B ⊔ D ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   u. cun 3119   (/)c0 3414   {csn 3583   X. cxp 4609   1oc1o 6388   ⊔ cdju 7014

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 4051  df-xp 4617  df-dju 7015

This theorem is referenced by:  djueq1  7017  djueq2  7018  casef  7065