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Theorem djueq12 6722
Description: Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
Assertion
Ref Expression
djueq12 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))

Proof of Theorem djueq12
StepHypRef Expression
1 xpeq2 4451 . . . 4 (𝐴 = 𝐵 → ({∅} × 𝐴) = ({∅} × 𝐵))
21adantr 270 . . 3 ((𝐴 = 𝐵𝐶 = 𝐷) → ({∅} × 𝐴) = ({∅} × 𝐵))
3 xpeq2 4451 . . . 4 (𝐶 = 𝐷 → ({1𝑜} × 𝐶) = ({1𝑜} × 𝐷))
43adantl 271 . . 3 ((𝐴 = 𝐵𝐶 = 𝐷) → ({1𝑜} × 𝐶) = ({1𝑜} × 𝐷))
52, 4uneq12d 3155 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (({∅} × 𝐴) ∪ ({1𝑜} × 𝐶)) = (({∅} × 𝐵) ∪ ({1𝑜} × 𝐷)))
6 df-dju 6721 . 2 (𝐴𝐶) = (({∅} × 𝐴) ∪ ({1𝑜} × 𝐶))
7 df-dju 6721 . 2 (𝐵𝐷) = (({∅} × 𝐵) ∪ ({1𝑜} × 𝐷))
85, 6, 73eqtr4g 2145 1 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  cun 2997  c0 3286  {csn 3444   × cxp 4434  1𝑜c1o 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
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