Theorem nfdju 7038

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 7034	. 2 |- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
2		nfcv 2319	. . . 4 |- F/_ x { (/) }
3		nfdju.1	. . . 4 |- F/_ x A
4	2, 3	nfxp 4652	. . 3 |- F/_ x ( { (/) } X. A )
5		nfcv 2319	. . . 4 |- F/_ x { 1o }
6		nfdju.2	. . . 4 |- F/_ x B
7	5, 6	nfxp 4652	. . 3 |- F/_ x ( { 1o } X. B )
8	4, 7	nfun 3291	. 2 |- F/_ x ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
9	1, 8	nfcxfr 2316	1 |- F/_ x ( A ⊔ B )

Syntax hints:   F/_wnfc 2306   u. cun 3127   (/)c0 3422   {csn 3592   X. cxp 4623   1oc1o 6407   ⊔ cdju 7033

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-un 3133  df-opab 4064  df-xp 4631  df-dju 7034

This theorem is referenced by:  ctiunctal  12434