Theorem bdsep2 14723

Description: Version of ax-bdsep 14721 with one disjoint variable condition removed and without initial universal quantifier. Use bdsep1 14722 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 2241	. . . . . 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 1824	. . 3 |- ( y = a -> ( A. x ( x e. b <-> ( x e. y /\ ph ) ) <-> A. x ( x e. b <-> ( x e. a /\ ph ) ) ) )
5	4	exbidv 1825	. 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 14722	. 2 |- E. b A. x ( x e. b <-> ( x e. y /\ ph ) )
8	5, 7	chvarv 1937	1 |- E. b A. x ( x e. b <-> ( x e. a /\ ph ) )

Syntax hints:   /\ wa 104   <-> wb 105   A.wal 1351   E.wex 1492  BOUNDED wbd 14649

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-ext 2159  ax-bdsep 14721

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-cleq 2170  df-clel 2173

This theorem is referenced by:  bdsepnft  14724  bdsepnfALT  14726