Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bdsepnf Unicode version

Theorem bdsepnf 13663
Description: Version of ax-bdsep 13659 with one disjoint variable condition removed, the other disjoint variable condition replaced by a nonfreeness hypothesis, and without initial universal quantifier. See also bdsepnfALT 13664. Use bdsep1 13660 when sufficient. (Contributed by BJ, 5-Oct-2019.)
Hypotheses
Ref Expression
bdsepnf.nf  |-  F/ b
ph
bdsepnf.1  |- BOUNDED  ph
Assertion
Ref Expression
bdsepnf  |-  E. b A. x ( x  e.  b  <->  ( x  e.  a  /\  ph )
)
Distinct variable group:    a, b, x
Allowed substitution hints:    ph( x, a, b)

Proof of Theorem bdsepnf
StepHypRef Expression
1 bdsepnf.1 . . 3  |- BOUNDED  ph
21bdsepnft 13662 . 2  |-  ( A. x F/ b ph  ->  E. b A. x ( x  e.  b  <->  ( x  e.  a  /\  ph )
) )
3 bdsepnf.nf . 2  |-  F/ b
ph
42, 3mpg 1438 1  |-  E. b A. x ( x  e.  b  <->  ( x  e.  a  /\  ph )
)
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104   A.wal 1340   F/wnf 1447   E.wex 1479  BOUNDED wbd 13587
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-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-bdsep 13659
This theorem depends on definitions:  df-bi 116  df-nf 1448  df-cleq 2157  df-clel 2160
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator