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Theorem nfdisjv 3785
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 3775 . 2  |-  (Disj  x  e.  A  B  <->  A. z E* x ( x  e.  A  /\  z  e.  B ) )
2 nfcv 2194 . . . . . 6  |-  F/_ y
x
3 nfdisjv.1 . . . . . 6  |-  F/_ y A
42, 3nfel 2202 . . . . 5  |-  F/ y  x  e.  A
5 nfdisjv.2 . . . . . 6  |-  F/_ y B
65nfcri 2188 . . . . 5  |-  F/ y  z  e.  B
74, 6nfan 1473 . . . 4  |-  F/ y ( x  e.  A  /\  z  e.  B
)
87nfmo 1936 . . 3  |-  F/ y E* x ( x  e.  A  /\  z  e.  B )
98nfal 1484 . 2  |-  F/ y A. z E* x
( x  e.  A  /\  z  e.  B
)
101, 9nfxfr 1379 1  |-  F/ yDisj  x  e.  A  B
Colors of variables: wff set class
Syntax hints:    /\ wa 101   A.wal 1257   F/wnf 1365    e. wcel 1409   E*wmo 1917   F/_wnfc 2181  Disj wdisj 3773
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rmo 2331  df-disj 3774
This theorem is referenced by: (None)
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