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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
StepHypRef 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
42, 3nfxp 4652 . . 3  |-  F/_ x
( { (/) }  X.  A )
5 nfcv 2319 . . . 4  |-  F/_ x { 1o }
6 nfdju.2 . . . 4  |-  F/_ x B
75, 6nfxp 4652 . . 3  |-  F/_ x
( { 1o }  X.  B )
84, 7nfun 3291 . 2  |-  F/_ x
( ( { (/) }  X.  A )  u.  ( { 1o }  X.  B ) )
91, 8nfcxfr 2316 1  |-  F/_ x
( A B )
Colors of variables: wff set class
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
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