Theorem axsep2 4108

Description: A less restrictive version of the Separation Scheme ax-sep 4107, where variables x and z can both appear free in the wff ph, which can therefore be thought of as ph ( x , z ). This version was derived from the more restrictive ax-sep 4107 with no additional set theory axioms. (Contributed by NM, 10-Dec-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Assertion

Ref	Expression
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 axsep2

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2234	. . . . . . 7 |- ( w = z -> ( x e. w <-> x e. z ) )
2	1	anbi1d 462	. . . . . 6 |- ( w = z -> ( ( x e. w /\ ( x e. z /\ ph ) ) <-> ( x e. z /\ ( x e. z /\ ph ) ) ) )
3		anabs5 568	. . . . . 6 |- ( ( x e. z /\ ( x e. z /\ ph ) ) <-> ( x e. z /\ ph ) )
4	2, 3	bitrdi 195	. . . . 5 |- ( w = z -> ( ( x e. w /\ ( x e. z /\ ph ) ) <-> ( x e. z /\ ph ) ) )
5	4	bibi2d 231	. . . 4 |- ( w = z -> ( ( x e. y <-> ( x e. w /\ ( x e. z /\ ph ) ) ) <-> ( x e. y <-> ( x e. z /\ ph ) ) ) )
6	5	albidv 1817	. . 3 |- ( w = z -> ( A. x ( x e. y <-> ( x e. w /\ ( x e. z /\ ph ) ) ) <-> A. x ( x e. y <-> ( x e. z /\ ph ) ) ) )
7	6	exbidv 1818	. 2 |- ( w = z -> ( E. y A. x ( x e. y <-> ( x e. w /\ ( x e. z /\ ph ) ) ) <-> E. y A. x ( x e. y <-> ( x e. z /\ ph ) ) ) )
8		ax-sep 4107	. 2 |- E. y A. x ( x e. y <-> ( x e. w /\ ( x e. z /\ ph ) ) )
9	7, 8	chvarv 1930	1 |- E. y A. x ( x e. y <-> ( x e. z /\ ph ) )

Syntax hints:   /\ wa 103   <-> wb 104   A.wal 1346   E.wex 1485

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-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152  ax-sep 4107

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-cleq 2163  df-clel 2166

This theorem is referenced by: (None)