Theorem djueq12 7038

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 4642	. . . 4 ⊢ (𝐴 = 𝐵 → ({∅} × 𝐴) = ({∅} × 𝐵))
2	1	adantr 276	. . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → ({∅} × 𝐴) = ({∅} × 𝐵))
3		xpeq2 4642	. . . 4 ⊢ (𝐶 = 𝐷 → ({1o} × 𝐶) = ({1o} × 𝐷))
4	3	adantl 277	. . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → ({1o} × 𝐶) = ({1o} × 𝐷))
5	2, 4	uneq12d 3291	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (({∅} × 𝐴) ∪ ({1o} × 𝐶)) = (({∅} × 𝐵) ∪ ({1o} × 𝐷)))
6		df-dju 7037	. 2 ⊢ (𝐴 ⊔ 𝐶) = (({∅} × 𝐴) ∪ ({1o} × 𝐶))
7		df-dju 7037	. 2 ⊢ (𝐵 ⊔ 𝐷) = (({∅} × 𝐵) ∪ ({1o} × 𝐷))
8	5, 6, 7	3eqtr4g 2235	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⊔ 𝐶) = (𝐵 ⊔ 𝐷))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353   ∪ cun 3128  ∅c0 3423  {csn 3593   × cxp 4625  1_oc1o 6410   ⊔ cdju 7036

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-un 3134  df-opab 4066  df-xp 4633  df-dju 7037

This theorem is referenced by:  djueq1  7039  djueq2  7040  casef  7087