Theorem nfdju 9362

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 9356	. 2 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
2		nfcv 2920	. . . 4 ⊢ Ⅎ𝑥{∅}
3		nfdju.1	. . . 4 ⊢ Ⅎ𝑥𝐴
4	2, 3	nfxp 5558	. . 3 ⊢ Ⅎ𝑥({∅} × 𝐴)
5		nfcv 2920	. . . 4 ⊢ Ⅎ𝑥{1o}
6		nfdju.2	. . . 4 ⊢ Ⅎ𝑥𝐵
7	5, 6	nfxp 5558	. . 3 ⊢ Ⅎ𝑥({1o} × 𝐵)
8	4, 7	nfun 4071	. 2 ⊢ Ⅎ𝑥(({∅} × 𝐴) ∪ ({1o} × 𝐵))
9	1, 8	nfcxfr 2918	1 ⊢ Ⅎ𝑥(𝐴 ⊔ 𝐵)

Syntax hints:  Ⅎwnfc 2900   ∪ cun 3857  ∅c0 4226  {csn 4523   × cxp 5523  1_oc1o 8106   ⊔ cdju 9353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-un 3864  df-opab 5096  df-xp 5531  df-dju 9356

This theorem is referenced by: (None)