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Theorem nfdisjv 3921
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 3911 . 2  |-  (Disj  x  e.  A  B  <->  A. z E* x ( x  e.  A  /\  z  e.  B ) )
2 nfcv 2281 . . . . . 6  |-  F/_ y
x
3 nfdisjv.1 . . . . . 6  |-  F/_ y A
42, 3nfel 2290 . . . . 5  |-  F/ y  x  e.  A
5 nfdisjv.2 . . . . . 6  |-  F/_ y B
65nfcri 2275 . . . . 5  |-  F/ y  z  e.  B
74, 6nfan 1544 . . . 4  |-  F/ y ( x  e.  A  /\  z  e.  B
)
87nfmo 2019 . . 3  |-  F/ y E* x ( x  e.  A  /\  z  e.  B )
98nfal 1555 . 2  |-  F/ y A. z E* x
( x  e.  A  /\  z  e.  B
)
101, 9nfxfr 1450 1  |-  F/ yDisj  x  e.  A  B
Colors of variables: wff set class
Syntax hints:    /\ wa 103   A.wal 1329   F/wnf 1436    e. wcel 1480   E*wmo 2000   F/_wnfc 2268  Disj wdisj 3909
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rmo 2424  df-disj 3910
This theorem is referenced by: (None)
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