Theorem nfdju 7007

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

Hypotheses

Ref	Expression
nfdju.1	|- F/_ x A
nfdju.2	|- F/_ x B

Assertion

Ref	Expression
nfdju	|- F/_ x ( A ⊔ B )

Proof of Theorem nfdju

Step	Hyp	Ref	Expression
1		df-dju 7003	. 2 |- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
2		nfcv 2308	. . . 4 |- F/_ x { (/) }
3		nfdju.1	. . . 4 |- F/_ x A
4	2, 3	nfxp 4631	. . 3 |- F/_ x ( { (/) } X. A )
5		nfcv 2308	. . . 4 |- F/_ x { 1o }
6		nfdju.2	. . . 4 |- F/_ x B
7	5, 6	nfxp 4631	. . 3 |- F/_ x ( { 1o } X. B )
8	4, 7	nfun 3278	. 2 |- F/_ x ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
9	1, 8	nfcxfr 2305	1 |- F/_ x ( A ⊔ B )

Syntax hints:   F/_wnfc 2295   u. cun 3114   (/)c0 3409   {csn 3576   X. cxp 4602   1oc1o 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