Theorem nfdju 7019

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 7015	. 2 |- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
2		nfcv 2312	. . . 4 |- F/_ x { (/) }
3		nfdju.1	. . . 4 |- F/_ x A
4	2, 3	nfxp 4638	. . 3 |- F/_ x ( { (/) } X. A )
5		nfcv 2312	. . . 4 |- F/_ x { 1o }
6		nfdju.2	. . . 4 |- F/_ x B
7	5, 6	nfxp 4638	. . 3 |- F/_ x ( { 1o } X. B )
8	4, 7	nfun 3283	. 2 |- F/_ x ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
9	1, 8	nfcxfr 2309	1 |- F/_ x ( A ⊔ B )

Syntax hints:   F/_wnfc 2299   u. cun 3119   (/)c0 3414   {csn 3583   X. cxp 4609   1oc1o 6388   ⊔ cdju 7014

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-un 3125  df-opab 4051  df-xp 4617  df-dju 7015

This theorem is referenced by:  ctiunctal  12396