Theorem nfdju 6927

Description: Bound-variable hypothesis builder for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)

Hypotheses

Ref	Expression
nfdju.1	⊢ Ⅎ𝑥𝐴
nfdju.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
nfdju	⊢ Ⅎ𝑥(𝐴 ⊔ 𝐵)

Proof of Theorem nfdju

Step	Hyp	Ref	Expression
1		df-dju 6923	. 2 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
2		nfcv 2281	. . . 4 ⊢ Ⅎ𝑥{∅}
3		nfdju.1	. . . 4 ⊢ Ⅎ𝑥𝐴
4	2, 3	nfxp 4566	. . 3 ⊢ Ⅎ𝑥({∅} × 𝐴)
5		nfcv 2281	. . . 4 ⊢ Ⅎ𝑥{1o}
6		nfdju.2	. . . 4 ⊢ Ⅎ𝑥𝐵
7	5, 6	nfxp 4566	. . 3 ⊢ Ⅎ𝑥({1o} × 𝐵)
8	4, 7	nfun 3232	. 2 ⊢ Ⅎ𝑥(({∅} × 𝐴) ∪ ({1o} × 𝐵))
9	1, 8	nfcxfr 2278	1 ⊢ Ⅎ𝑥(𝐴 ⊔ 𝐵)

Syntax hints:  Ⅎwnfc 2268   ∪ cun 3069  ∅c0 3363  {csn 3527   × cxp 4537  1_oc1o 6306   ⊔ cdju 6922

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-un 3075  df-opab 3990  df-xp 4545  df-dju 6923

This theorem is referenced by: (None)