Theorem sepg 4246

Description: Version of the axiom of separation where the "containing set" is a class variable and the sethood assumption is in the antecedent. (Contributed by NM, 21-Jun-1993.) Put sepgi 4247 in closed form. (Revised by BJ, 2-Jul-2022.)

Assertion

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

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

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

Proof of Theorem sepg

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2302	. . . . . 6 |- ( z = A -> ( x e. z <-> x e. A ) )
2	1	anbi1d 469	. . . . 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 1877	. . 3 |- ( z = A -> ( A. x ( x e. y <-> ( x e. z /\ ph ) ) <-> A. x ( x e. y <-> ( x e. A /\ ph ) ) ) )
5	4	exbidv 1878	. 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		ax-sep 4244	. 2 |- E. y A. x ( x e. y <-> ( x e. z /\ ph ) )
7	5, 6	vtoclg 2883	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 1400   = wceq 1402   E.wex 1545   e. wcel 2209

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-sep 4244

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823

This theorem is referenced by:  sepgi  4247