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

Theorem nfdisjv 3994
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 3984 . 2  |-  (Disj  x  e.  A  B  <->  A. z E* x ( x  e.  A  /\  z  e.  B ) )
2 nfcv 2319 . . . . . 6  |-  F/_ y
x
3 nfdisjv.1 . . . . . 6  |-  F/_ y A
42, 3nfel 2328 . . . . 5  |-  F/ y  x  e.  A
5 nfdisjv.2 . . . . . 6  |-  F/_ y B
65nfcri 2313 . . . . 5  |-  F/ y  z  e.  B
74, 6nfan 1565 . . . 4  |-  F/ y ( x  e.  A  /\  z  e.  B
)
87nfmo 2046 . . 3  |-  F/ y E* x ( x  e.  A  /\  z  e.  B )
98nfal 1576 . 2  |-  F/ y A. z E* x
( x  e.  A  /\  z  e.  B
)
101, 9nfxfr 1474 1  |-  F/ yDisj  x  e.  A  B
Colors of variables: wff set class
Syntax hints:    /\ wa 104   A.wal 1351   F/wnf 1460   E*wmo 2027    e. wcel 2148   F/_wnfc 2306  Disj wdisj 3982
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-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rmo 2463  df-disj 3983
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator