Theorem bdsepnf 13923

Description: Version of ax-bdsep 13919 with one disjoint variable condition removed, the other disjoint variable condition replaced by a nonfreeness hypothesis, and without initial universal quantifier. See also bdsepnfALT 13924. Use bdsep1 13920 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 13922	. 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 1444	1 |- E. b A. x ( x e. b <-> ( x e. a /\ ph ) )

Syntax hints:   /\ wa 103   <-> wb 104   A.wal 1346   F/wnf 1453   E.wex 1485  BOUNDED wbd 13847

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-bdsep 13919

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-cleq 2163  df-clel 2166

This theorem is referenced by: (None)