Theorem nfdju 7301

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 7297	. 2 |- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
2		nfcv 2375	. . . 4 |- F/_ x { (/) }
3		nfdju.1	. . . 4 |- F/_ x A
4	2, 3	nfxp 4758	. . 3 |- F/_ x ( { (/) } X. A )
5		nfcv 2375	. . . 4 |- F/_ x { 1o }
6		nfdju.2	. . . 4 |- F/_ x B
7	5, 6	nfxp 4758	. . 3 |- F/_ x ( { 1o } X. B )
8	4, 7	nfun 3365	. 2 |- F/_ x ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
9	1, 8	nfcxfr 2372	1 |- F/_ x ( A ⊔ B )

Syntax hints:   F/_wnfc 2362   u. cun 3199   (/)c0 3496   {csn 3673   X. cxp 4729   1oc1o 6618   ⊔ cdju 7296

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-un 3205  df-opab 4156  df-xp 4737  df-dju 7297

This theorem is referenced by:  ctiunctal  13142