Theorem bdzfauscl 13088

Description: Closed form of the version of zfauscl 4048 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 2203	. . . . . 6 |- ( z = A -> ( x e. z <-> x e. A ) )
2	1	anbi1d 460	. . . . 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 1796	. . 3 |- ( z = A -> ( A. x ( x e. y <-> ( x e. z /\ ph ) ) <-> A. x ( x e. y <-> ( x e. A /\ ph ) ) ) )
5	4	exbidv 1797	. 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 13083	. 2 |- E. y A. x ( x e. y <-> ( x e. z /\ ph ) )
8	5, 7	vtoclg 2746	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 1329   = wceq 1331   E.wex 1468   e. wcel 1480  BOUNDED wbd 13010

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-bdsep 13082

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688

This theorem is referenced by:  bdinex1  13097