Theorem axsep2 3867

Description: A less restrictive version of the Separation Scheme ax-sep 3866, 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 3866 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.yA.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 2098	. . . . . . 7 |- (w = z -> (x e. w <-> x e. z))
2	1	anbi1d 438	. . . . . 6 |- (w = z -> ((x e. w /\ (x e. z /\ ph)) <-> (x e. z /\ (x e. z /\ ph))))
3		anabs5 507	. . . . . 6 |- ((x e. z /\ (x e. z /\ ph)) <-> (x e. z /\ ph))
4	2, 3	syl6bb 185	. . . . 5 |- (w = z -> ((x e. w /\ (x e. z /\ ph)) <-> (x e. z /\ ph)))
5	4	bibi2d 221	. . . 4 |- (w = z -> ((x e. y <-> (x e. w /\ (x e. z /\ ph))) <-> (x e. y <-> (x e. z /\ ph))))
6	5	albidv 1702	. . 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 1703	. 2 |- (w = z -> (E.yA.x(x e. y <-> (x e. w /\ (x e. z /\ ph))) <-> E.yA.x(x e. y <-> (x e. z /\ ph))))
8		ax-sep 3866	. 2 |- E.yA.x(x e. y <-> (x e. w /\ (x e. z /\ ph)))
9	7, 8	chvarv 1809	1 |- E.yA.x(x e. y <-> (x e. z /\ ph))

Syntax hints:   /\ wa 97   <-> wb 98  A.wal 1240  E.wex 1378

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019  ax-sep 3866

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-cleq 2030  df-clel 2033

This theorem is referenced by: (None)