Theorem bdzfauscl 13772

Description: Closed form of the version of zfauscl 4102 for bounded formulas using bounded separation. (Contributed by BJ, 13-Nov-2019.)

Hypothesis

Ref	Expression
bdzfauscl.bd	|- BOUNDED ph

Assertion

Ref	Expression
bdzfauscl	|- ( A e. V -> E. y A. x ( x e. y <-> ( x e. A /\ ph ) ) )

Distinct variable groups:   x, y, A   ph, y

Allowed substitution hints:   ph( x)   V( x, y)

Proof of Theorem bdzfauscl

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2230	. . . . . 6 |- ( z = A -> ( x e. z <-> x e. A ) )
2	1	anbi1d 461	. . . . 5 |- ( z = A -> ( ( x e. z /\ ph ) <-> ( x e. A /\ ph ) ) )
3	2	bibi2d 231	. . . 4 |- ( z = A -> ( ( x e. y <-> ( x e. z /\ ph ) ) <-> ( x e. y <-> ( x e. A /\ ph ) ) ) )
4	3	albidv 1812	. . 3 |- ( z = A -> ( A. x ( x e. y <-> ( x e. z /\ ph ) ) <-> A. x ( x e. y <-> ( x e. A /\ ph ) ) ) )
5	4	exbidv 1813	. 2 |- ( z = A -> ( E. y A. x ( x e. y <-> ( x e. z /\ ph ) ) <-> E. y A. x ( x e. y <-> ( x e. A /\ ph ) ) ) )
6		bdzfauscl.bd	. . 3 |- BOUNDED ph
7	6	bdsep1 13767	. 2 |- E. y A. x ( x e. y <-> ( x e. z /\ ph ) )
8	5, 7	vtoclg 2786	1 |- ( A e. V -> E. y A. x ( x e. y <-> ( x e. A /\ ph ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1341   = wceq 1343   E.wex 1480   e. wcel 2136  BOUNDED wbd 13694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-bdsep 13766

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728

This theorem is referenced by:  bdinex1  13781