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Theorem nfdju 9947
Description: Bound-variable hypothesis builder for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
Hypotheses
Ref Expression
nfdju.1 𝑥𝐴
nfdju.2 𝑥𝐵
Assertion
Ref Expression
nfdju 𝑥(𝐴𝐵)

Proof of Theorem nfdju
StepHypRef Expression
1 df-dju 9941 . 2 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
2 nfcv 2905 . . . 4 𝑥{∅}
3 nfdju.1 . . . 4 𝑥𝐴
42, 3nfxp 5718 . . 3 𝑥({∅} × 𝐴)
5 nfcv 2905 . . . 4 𝑥{1o}
6 nfdju.2 . . . 4 𝑥𝐵
75, 6nfxp 5718 . . 3 𝑥({1o} × 𝐵)
84, 7nfun 4170 . 2 𝑥(({∅} × 𝐴) ∪ ({1o} × 𝐵))
91, 8nfcxfr 2903 1 𝑥(𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wnfc 2890  cun 3949  c0 4333  {csn 4626   × cxp 5683  1oc1o 8499  cdju 9938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-v 3482  df-un 3956  df-opab 5206  df-xp 5691  df-dju 9941
This theorem is referenced by: (None)
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