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Theorem djueq12 7016
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 4626 . . . 4 (𝐴 = 𝐵 → ({∅} × 𝐴) = ({∅} × 𝐵))
21adantr 274 . . 3 ((𝐴 = 𝐵𝐶 = 𝐷) → ({∅} × 𝐴) = ({∅} × 𝐵))
3 xpeq2 4626 . . . 4 (𝐶 = 𝐷 → ({1o} × 𝐶) = ({1o} × 𝐷))
43adantl 275 . . 3 ((𝐴 = 𝐵𝐶 = 𝐷) → ({1o} × 𝐶) = ({1o} × 𝐷))
52, 4uneq12d 3282 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (({∅} × 𝐴) ∪ ({1o} × 𝐶)) = (({∅} × 𝐵) ∪ ({1o} × 𝐷)))
6 df-dju 7015 . 2 (𝐴𝐶) = (({∅} × 𝐴) ∪ ({1o} × 𝐶))
7 df-dju 7015 . 2 (𝐵𝐷) = (({∅} × 𝐵) ∪ ({1o} × 𝐷))
85, 6, 73eqtr4g 2228 1 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1348  cun 3119  c0 3414  {csn 3583   × 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
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