Theorem djueq12 6722

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 4451	. . . 4 ⊢ (𝐴 = 𝐵 → ({∅} × 𝐴) = ({∅} × 𝐵))
2	1	adantr 270	. . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → ({∅} × 𝐴) = ({∅} × 𝐵))
3		xpeq2 4451	. . . 4 ⊢ (𝐶 = 𝐷 → ({1𝑜} × 𝐶) = ({1𝑜} × 𝐷))
4	3	adantl 271	. . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → ({1𝑜} × 𝐶) = ({1𝑜} × 𝐷))
5	2, 4	uneq12d 3155	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (({∅} × 𝐴) ∪ ({1𝑜} × 𝐶)) = (({∅} × 𝐵) ∪ ({1𝑜} × 𝐷)))
6		df-dju 6721	. 2 ⊢ (𝐴 ⊔ 𝐶) = (({∅} × 𝐴) ∪ ({1𝑜} × 𝐶))
7		df-dju 6721	. 2 ⊢ (𝐵 ⊔ 𝐷) = (({∅} × 𝐵) ∪ ({1𝑜} × 𝐷))
8	5, 6, 7	3eqtr4g 2145	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⊔ 𝐶) = (𝐵 ⊔ 𝐷))

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1289   ∪ cun 2997  ∅c0 3286  {csn 3444   × cxp 4434  1_𝑜c1o 6166   ⊔ cdju 6720

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-opab 3898  df-xp 4442  df-dju 6721

This theorem is referenced by:  djueq1  6723  djueq2  6724  casef  6769