Theorem bdsep1 10005

Description: Version of ax-bdsep 10004 without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.)

Hypothesis

Ref	Expression
bdsep1.1	|- BOUNDED ph

Assertion

Ref	Expression
bdsep1	|- E. b A. x ( x e. b <-> ( x e. a /\ ph ) )

Distinct variable groups:   a, b, x   ph, a, b

Allowed substitution hint:   ph( x)

Proof of Theorem bdsep1

Step	Hyp	Ref	Expression
1		bdsep1.1	. . 3 |- BOUNDED ph
2	1	ax-bdsep 10004	. 2 |- A. a E. b A. x ( x e. b <-> ( x e. a /\ ph ) )
3	2	spi 1429	1 |- E. b A. x ( x e. b <-> ( x e. a /\ ph ) )

Syntax hints:   /\ wa 97   <-> wb 98   A.wal 1241   E.wex 1381  BOUNDED wbd 9932

This theorem was proved from axioms:  ax-mp 7  ax-4 1400  ax-bdsep 10004

This theorem is referenced by:  bdsep2  10006  bdzfauscl  10010  bdbm1.3ii  10011  bj-axemptylem  10012  bj-nalset  10015