Theorem bdsep1 16923

Description: Version of ax-bdsep 16922 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 16922	. 2 |- A. a E. b A. x ( x e. b <-> ( x e. a /\ ph ) )
3	2	spi 1589	1 |- E. b A. x ( x e. b <-> ( x e. a /\ ph ) )

This proof depends on syntax axioms:   /\ wa 104   <-> wb 105   A.wal 1400   E.wex 1545  BOUNDED wbd 16850

This proof depends on axioms:  ax-mp 5  ax-4 1563  ax-bdsep 16922

This theorem is used by:  bdsep2  16924  bdsepg  16928  bdbm1.3ii  16929  bj-axemptylem  16930  bj-nalset  16933