ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nfdisjv Unicode version

Theorem nfdisjv 3856
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 3846 . 2  |-  (Disj  x  e.  A  B  <->  A. z E* x ( x  e.  A  /\  z  e.  B ) )
2 nfcv 2235 . . . . . 6  |-  F/_ y
x
3 nfdisjv.1 . . . . . 6  |-  F/_ y A
42, 3nfel 2244 . . . . 5  |-  F/ y  x  e.  A
5 nfdisjv.2 . . . . . 6  |-  F/_ y B
65nfcri 2229 . . . . 5  |-  F/ y  z  e.  B
74, 6nfan 1509 . . . 4  |-  F/ y ( x  e.  A  /\  z  e.  B
)
87nfmo 1975 . . 3  |-  F/ y E* x ( x  e.  A  /\  z  e.  B )
98nfal 1520 . 2  |-  F/ y A. z E* x
( x  e.  A  /\  z  e.  B
)
101, 9nfxfr 1415 1  |-  F/ yDisj  x  e.  A  B
Colors of variables: wff set class
Syntax hints:    /\ wa 103   A.wal 1294   F/wnf 1401    e. wcel 1445   E*wmo 1956   F/_wnfc 2222  Disj wdisj 3844
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 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-cleq 2088  df-clel 2091  df-nfc 2224  df-rmo 2378  df-disj 3845
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator