Theorem bdsep1 10943

Description: Version of ax-bdsep 10942 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 10942	. 2 |- A. a E. b A. x ( x e. b <-> ( x e. a /\ ph ) )
3	2	spi 1470	1 |- E. b A. x ( x e. b <-> ( x e. a /\ ph ) )

Syntax hints:   /\ wa 102   <-> wb 103   A.wal 1283   E.wex 1422  BOUNDED wbd 10870

This theorem was proved from axioms:  ax-mp 7  ax-4 1441  ax-bdsep 10942

This theorem is referenced by:  bdsep2  10944  bdzfauscl  10948  bdbm1.3ii  10949  bj-axemptylem  10950  bj-nalset  10953