Theorem nfdju 7101

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 7097	. 2 |- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
2		nfcv 2336	. . . 4 |- F/_ x { (/) }
3		nfdju.1	. . . 4 |- F/_ x A
4	2, 3	nfxp 4686	. . 3 |- F/_ x ( { (/) } X. A )
5		nfcv 2336	. . . 4 |- F/_ x { 1o }
6		nfdju.2	. . . 4 |- F/_ x B
7	5, 6	nfxp 4686	. . 3 |- F/_ x ( { 1o } X. B )
8	4, 7	nfun 3315	. 2 |- F/_ x ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
9	1, 8	nfcxfr 2333	1 |- F/_ x ( A ⊔ B )

Syntax hints:   F/_wnfc 2323   u. cun 3151   (/)c0 3446   {csn 3618   X. cxp 4657   1oc1o 6462   ⊔ cdju 7096

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-un 3157  df-opab 4091  df-xp 4665  df-dju 7097

This theorem is referenced by:  ctiunctal  12598