Theorem bdsep2 10677

Description: Version of ax-bdsep 10675 with one DV condition removed and without initial universal quantifier. Use bdsep1 10676 when sufficient. (Contributed by BJ, 5-Oct-2019.)

Hypothesis

Ref	Expression
bdsep2.1	|- BOUNDED ph

Assertion

Ref	Expression
bdsep2	|- E. b A. x ( x e. b <-> ( x e. a /\ ph ) )

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

Allowed substitution hints:   ph( x, a)

Proof of Theorem bdsep2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2142	. . . . . 6 |- ( y = a -> ( x e. y <-> x e. a ) )
2	1	anbi1d 452	. . . . 5 |- ( y = a -> ( ( x e. y /\ ph ) <-> ( x e. a /\ ph ) ) )
3	2	bibi2d 230	. . . 4 |- ( y = a -> ( ( x e. b <-> ( x e. y /\ ph ) ) <-> ( x e. b <-> ( x e. a /\ ph ) ) ) )
4	3	albidv 1745	. . 3 |- ( y = a -> ( A. x ( x e. b <-> ( x e. y /\ ph ) ) <-> A. x ( x e. b <-> ( x e. a /\ ph ) ) ) )
5	4	exbidv 1746	. 2 |- ( y = a -> ( E. b A. x ( x e. b <-> ( x e. y /\ ph ) ) <-> E. b A. x ( x e. b <-> ( x e. a /\ ph ) ) ) )
6		bdsep2.1	. . 3 |- BOUNDED ph
7	6	bdsep1 10676	. 2 |- E. b A. x ( x e. b <-> ( x e. y /\ ph ) )
8	5, 7	chvarv 1853	1 |- E. b A. x ( x e. b <-> ( x e. a /\ ph ) )

Syntax hints:   /\ wa 102   <-> wb 103   A.wal 1282   E.wex 1421  BOUNDED wbd 10603

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-ext 2063  ax-bdsep 10675

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-cleq 2074  df-clel 2077

This theorem is referenced by:  bdsepnft  10678  bdsepnfALT  10680