Theorem bdsepnf 13257

Description: Version of ax-bdsep 13253 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 13258. Use bdsep1 13254 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

Step	Hyp	Ref	Expression
1		bdsepnf.1	. . 3 |- BOUNDED ph
2	1	bdsepnft 13256	. 2 |- ( A. x F/ b ph -> E. b A. x ( x e. b <-> ( x e. a /\ ph ) ) )
3		bdsepnf.nf	. 2 |- F/ b ph
4	2, 3	mpg 1428	1 |- E. b A. x ( x e. b <-> ( x e. a /\ ph ) )

Syntax hints:   /\ wa 103   <-> wb 104   A.wal 1330   F/wnf 1437   E.wex 1469  BOUNDED wbd 13181

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-bdsep 13253

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-cleq 2133  df-clel 2136

This theorem is referenced by: (None)