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Theorem nfdju 9899
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 9893 . 2 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
2 nfcv 2895 . . . 4 𝑥{∅}
3 nfdju.1 . . . 4 𝑥𝐴
42, 3nfxp 5700 . . 3 𝑥({∅} × 𝐴)
5 nfcv 2895 . . . 4 𝑥{1o}
6 nfdju.2 . . . 4 𝑥𝐵
75, 6nfxp 5700 . . 3 𝑥({1o} × 𝐵)
84, 7nfun 4158 . 2 𝑥(({∅} × 𝐴) ∪ ({1o} × 𝐵))
91, 8nfcxfr 2893 1 𝑥(𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wnfc 2875  cun 3939  c0 4315  {csn 4621   × cxp 5665  1oc1o 8455  cdju 9890
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-un 3946  df-opab 5202  df-xp 5673  df-dju 9893
This theorem is referenced by: (None)
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