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

Theorem bdsepnf 13116
Description: Version of ax-bdsep 13112 with one disjoint variable condition removed, the other disjoint variable condition replaced by a non-freeness hypothesis, and without initial universal quantifier. See also bdsepnfALT 13117. Use bdsep1 13113 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 13115 . 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 1427 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 1329   F/wnf 1436   E.wex 1468  BOUNDED wbd 13040
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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-bdsep 13112
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-cleq 2132  df-clel 2135
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator