Theorem bdsep2 15378

Description: Version of ax-bdsep 15376 with one disjoint variable condition removed and without initial universal quantifier. Use bdsep1 15377 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 2257	. . . . . 6 |- ( y = a -> ( x e. y <-> x e. a ) )
2	1	anbi1d 465	. . . . 5 |- ( y = a -> ( ( x e. y /\ ph ) <-> ( x e. a /\ ph ) ) )
3	2	bibi2d 232	. . . 4 |- ( y = a -> ( ( x e. b <-> ( x e. y /\ ph ) ) <-> ( x e. b <-> ( x e. a /\ ph ) ) ) )
4	3	albidv 1835	. . 3 |- ( y = a -> ( A. x ( x e. b <-> ( x e. y /\ ph ) ) <-> A. x ( x e. b <-> ( x e. a /\ ph ) ) ) )
5	4	exbidv 1836	. 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 15377	. 2 |- E. b A. x ( x e. b <-> ( x e. y /\ ph ) )
8	5, 7	chvarv 1953	1 |- E. b A. x ( x e. b <-> ( x e. a /\ ph ) )

Syntax hints:   /\ wa 104   <-> wb 105   A.wal 1362   E.wex 1503  BOUNDED wbd 15304

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-ext 2175  ax-bdsep 15376

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-cleq 2186  df-clel 2189

This theorem is referenced by:  bdsepnft  15379  bdsepnfALT  15381