Theorem bdsepnf 16023

Description: Version of ax-bdsep 16019 with one disjoint variable condition removed, the other disjoint variable condition replaced by a nonfreeness hypothesis, and without initial universal quantifier. See also bdsepnfALT 16024. Use bdsep1 16020 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 16022	. 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 1475	1 |- E. b A. x ( x e. b <-> ( x e. a /\ ph ) )

Syntax hints:   /\ wa 104   <-> wb 105   A.wal 1371   F/wnf 1484   E.wex 1516  BOUNDED wbd 15947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-bdsep 16019

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-cleq 2200  df-clel 2203

This theorem is referenced by: (None)