Theorem nfdju 7007

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 7003	. 2 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
2		nfcv 2308	. . . 4 ⊢ Ⅎ𝑥{∅}
3		nfdju.1	. . . 4 ⊢ Ⅎ𝑥𝐴
4	2, 3	nfxp 4631	. . 3 ⊢ Ⅎ𝑥({∅} × 𝐴)
5		nfcv 2308	. . . 4 ⊢ Ⅎ𝑥{1o}
6		nfdju.2	. . . 4 ⊢ Ⅎ𝑥𝐵
7	5, 6	nfxp 4631	. . 3 ⊢ Ⅎ𝑥({1o} × 𝐵)
8	4, 7	nfun 3278	. 2 ⊢ Ⅎ𝑥(({∅} × 𝐴) ∪ ({1o} × 𝐵))
9	1, 8	nfcxfr 2305	1 ⊢ Ⅎ𝑥(𝐴 ⊔ 𝐵)

Syntax hints:  Ⅎwnfc 2295   ∪ cun 3114  ∅c0 3409  {csn 3576   × cxp 4602  1_oc1o 6377   ⊔ cdju 7002

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-un 3120  df-opab 4044  df-xp 4610  df-dju 7003

This theorem is referenced by:  ctiunctal  12374