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Theorem nfdisjv 3840
Description: Bound-variable hypothesis builder for disjoint collection. (Contributed by Jim Kingdon, 19-Aug-2018.)
Hypotheses
Ref Expression
nfdisjv.1  |-  F/_ y A
nfdisjv.2  |-  F/_ y B
Assertion
Ref Expression
nfdisjv  |-  F/ yDisj  x  e.  A  B
Distinct variable group:    x, y
Allowed substitution hints:    A( x, y)    B( x, y)

Proof of Theorem nfdisjv
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 dfdisj2 3830 . 2  |-  (Disj  x  e.  A  B  <->  A. z E* x ( x  e.  A  /\  z  e.  B ) )
2 nfcv 2229 . . . . . 6  |-  F/_ y
x
3 nfdisjv.1 . . . . . 6  |-  F/_ y A
42, 3nfel 2238 . . . . 5  |-  F/ y  x  e.  A
5 nfdisjv.2 . . . . . 6  |-  F/_ y B
65nfcri 2223 . . . . 5  |-  F/ y  z  e.  B
74, 6nfan 1503 . . . 4  |-  F/ y ( x  e.  A  /\  z  e.  B
)
87nfmo 1969 . . 3  |-  F/ y E* x ( x  e.  A  /\  z  e.  B )
98nfal 1514 . 2  |-  F/ y A. z E* x
( x  e.  A  /\  z  e.  B
)
101, 9nfxfr 1409 1  |-  F/ yDisj  x  e.  A  B
Colors of variables: wff set class
Syntax hints:    /\ wa 103   A.wal 1288   F/wnf 1395    e. wcel 1439   E*wmo 1950   F/_wnfc 2216  Disj wdisj 3828
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rmo 2368  df-disj 3829
This theorem is referenced by: (None)
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