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Theorem djueq12 6890
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 4522 . . . 4 (𝐴 = 𝐵 → ({∅} × 𝐴) = ({∅} × 𝐵))
21adantr 272 . . 3 ((𝐴 = 𝐵𝐶 = 𝐷) → ({∅} × 𝐴) = ({∅} × 𝐵))
3 xpeq2 4522 . . . 4 (𝐶 = 𝐷 → ({1o} × 𝐶) = ({1o} × 𝐷))
43adantl 273 . . 3 ((𝐴 = 𝐵𝐶 = 𝐷) → ({1o} × 𝐶) = ({1o} × 𝐷))
52, 4uneq12d 3199 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (({∅} × 𝐴) ∪ ({1o} × 𝐶)) = (({∅} × 𝐵) ∪ ({1o} × 𝐷)))
6 df-dju 6889 . 2 (𝐴𝐶) = (({∅} × 𝐴) ∪ ({1o} × 𝐶))
7 df-dju 6889 . 2 (𝐵𝐷) = (({∅} × 𝐵) ∪ ({1o} × 𝐷))
85, 6, 73eqtr4g 2173 1 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1314  cun 3037  c0 3331  {csn 3495   × cxp 4505  1oc1o 6272  cdju 6888
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-opab 3958  df-xp 4513  df-dju 6889
This theorem is referenced by:  djueq1  6891  djueq2  6892  casef  6939
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