ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nfdju GIF version

Theorem nfdju 6934
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 6930 . 2 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
2 nfcv 2282 . . . 4 𝑥{∅}
3 nfdju.1 . . . 4 𝑥𝐴
42, 3nfxp 4573 . . 3 𝑥({∅} × 𝐴)
5 nfcv 2282 . . . 4 𝑥{1o}
6 nfdju.2 . . . 4 𝑥𝐵
75, 6nfxp 4573 . . 3 𝑥({1o} × 𝐵)
84, 7nfun 3236 . 2 𝑥(({∅} × 𝐴) ∪ ({1o} × 𝐵))
91, 8nfcxfr 2279 1 𝑥(𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wnfc 2269  cun 3073  c0 3367  {csn 3531   × cxp 4544  1oc1o 6313  cdju 6929
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-un 3079  df-opab 3997  df-xp 4552  df-dju 6930
This theorem is referenced by:  ctiunctal  11988
  Copyright terms: Public domain W3C validator