Theorem nfdju 7372

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 7368	. 2 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
2		nfcv 2392	. . . 4 ⊢ Ⅎ𝑥{∅}
3		nfdju.1	. . . 4 ⊢ Ⅎ𝑥𝐴
4	2, 3	nfxp 4796	. . 3 ⊢ Ⅎ𝑥({∅} × 𝐴)
5		nfcv 2392	. . . 4 ⊢ Ⅎ𝑥{1o}
6		nfdju.2	. . . 4 ⊢ Ⅎ𝑥𝐵
7	5, 6	nfxp 4796	. . 3 ⊢ Ⅎ𝑥({1o} × 𝐵)
8	4, 7	nfun 3385	. 2 ⊢ Ⅎ𝑥(({∅} × 𝐴) ∪ ({1o} × 𝐵))
9	1, 8	nfcxfr 2389	1 ⊢ Ⅎ𝑥(𝐴 ⊔ 𝐵)

Syntax hints:  Ⅎwnfc 2379   ∪ cun 3218  ∅c0 3520  {csn 3705   × cxp 4767  1_oc1o 6670   ⊔ cdju 7367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-un 3224  df-opab 4188  df-xp 4775  df-dju 7368

This theorem is referenced by:  ctiunctal  13310