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Theorem djueq12 9125
 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 5424 . . . 4 (𝐴 = 𝐵 → ({∅} × 𝐴) = ({∅} × 𝐵))
21adantr 473 . . 3 ((𝐴 = 𝐵𝐶 = 𝐷) → ({∅} × 𝐴) = ({∅} × 𝐵))
3 xpeq2 5424 . . . 4 (𝐶 = 𝐷 → ({1o} × 𝐶) = ({1o} × 𝐷))
43adantl 474 . . 3 ((𝐴 = 𝐵𝐶 = 𝐷) → ({1o} × 𝐶) = ({1o} × 𝐷))
52, 4uneq12d 4022 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (({∅} × 𝐴) ∪ ({1o} × 𝐶)) = (({∅} × 𝐵) ∪ ({1o} × 𝐷)))
6 df-dju 9122 . 2 (𝐴𝐶) = (({∅} × 𝐴) ∪ ({1o} × 𝐶))
7 df-dju 9122 . 2 (𝐵𝐷) = (({∅} × 𝐵) ∪ ({1o} × 𝐷))
85, 6, 73eqtr4g 2832 1 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 387   = wceq 1508   ∪ cun 3820  ∅c0 4172  {csn 4435   × cxp 5401  1oc1o 7896   ⊔ cdju 9119 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2743 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-v 3410  df-un 3827  df-opab 4988  df-xp 5409  df-dju 9122 This theorem is referenced by:  djueq1  9126  djueq2  9127  isfin5  9517  alephadd  9795
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