Theorem bdzfauscl 16025

Description: Closed form of the version of zfauscl 4180 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 2271	. . . . . 6 |- ( z = A -> ( x e. z <-> x e. A ) )
2	1	anbi1d 465	. . . . 5 |- ( z = A -> ( ( x e. z /\ ph ) <-> ( x e. A /\ ph ) ) )
3	2	bibi2d 232	. . . 4 |- ( z = A -> ( ( x e. y <-> ( x e. z /\ ph ) ) <-> ( x e. y <-> ( x e. A /\ ph ) ) ) )
4	3	albidv 1848	. . 3 |- ( z = A -> ( A. x ( x e. y <-> ( x e. z /\ ph ) ) <-> A. x ( x e. y <-> ( x e. A /\ ph ) ) ) )
5	4	exbidv 1849	. 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 16020	. 2 |- E. y A. x ( x e. y <-> ( x e. z /\ ph ) )
8	5, 7	vtoclg 2838	1 |- ( A e. V -> E. y A. x ( x e. y <-> ( x e. A /\ ph ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1371   = wceq 1373   E.wex 1516   e. wcel 2178  BOUNDED wbd 15947

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-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189  ax-bdsep 16019

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778

This theorem is referenced by:  bdinex1  16034