Theorem bj-axsep2 32921

Description: Remove dependency on ax-12 2047 and ax-13 2246 from axsep2 4782 while shortening its proof. Remark: the comment in zfauscl 4783 is misleading: the essential use of ax-ext 2602 is the one via eleq2 2690 and not the one via vtocl 3259, since the latter can be proved without ax-ext 2602 (see bj-vtocl 32909). (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-axsep2	|- E. y A. x ( x e. y <-> ( x e. z /\ ph ) )

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

Allowed substitution hints:   ph( x, z)

Proof of Theorem bj-axsep2

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elequ2 2004	. . . . . 6 |- ( w = z -> ( x e. w <-> x e. z ) )
2	1	anbi1d 741	. . . . 5 |- ( w = z -> ( ( x e. w /\ ph ) <-> ( x e. z /\ ph ) ) )
3	2	bibi2d 332	. . . 4 |- ( w = z -> ( ( x e. y <-> ( x e. w /\ ph ) ) <-> ( x e. y <-> ( x e. z /\ ph ) ) ) )
4	3	albidv 1849	. . 3 |- ( w = z -> ( A. x ( x e. y <-> ( x e. w /\ ph ) ) <-> A. x ( x e. y <-> ( x e. z /\ ph ) ) ) )
5	4	exbidv 1850	. 2 |- ( w = z -> ( E. y A. x ( x e. y <-> ( x e. w /\ ph ) ) <-> E. y A. x ( x e. y <-> ( x e. z /\ ph ) ) ) )
6		ax-sep 4781	. 2 |- E. y A. x ( x e. y <-> ( x e. w /\ ph ) )
7	5, 6	bj-chvarvv 32726	1 |- E. y A. x ( x e. y <-> ( x e. z /\ ph ) )

Syntax hints:   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by: (None)