Theorem axsep2 3439

Description: A less restrictive version of the Separation Scheme axsep 3437, 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 3438 with no additional set theory axioms.

Assertion

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

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

Proof of Theorem axsep2

Step	Hyp	Ref	Expression
1		a9e 1483	. 2 |- E.w w = z
2		ax-sep 3438	. . . 4 |- E.yA.x(x e. y <-> (x e. w /\ (x e. z /\ ph)))
3		elequ2 1497	. . . . . . . . . . 11 |- (w = z -> (x e. w <-> x e. z))
4	3	biimprd 171	. . . . . . . . . 10 |- (w = z -> (x e. z -> x e. w))
5	4	pm4.71rd 701	. . . . . . . . 9 |- (w = z -> (x e. z <-> (x e. w /\ x e. z)))
6	5	anbi1d 679	. . . . . . . 8 |- (w = z -> ((x e. z /\ ph) <-> ((x e. w /\ x e. z) /\ ph)))
7		anass 487	. . . . . . . 8 |- (((x e. w /\ x e. z) /\ ph) <-> (x e. w /\ (x e. z /\ ph)))
8	6, 7	syl6bb 595	. . . . . . 7 |- (w = z -> ((x e. z /\ ph) <-> (x e. w /\ (x e. z /\ ph))))
9	8	bibi2d 680	. . . . . 6 |- (w = z -> ((x e. y <-> (x e. z /\ ph)) <-> (x e. y <-> (x e. w /\ (x e. z /\ ph)))))
10	9	albidv 1656	. . . . 5 |- (w = z -> (A.x(x e. y <-> (x e. z /\ ph)) <-> A.x(x e. y <-> (x e. w /\ (x e. z /\ ph)))))
11	10	exbidv 1657	. . . 4 |- (w = z -> (E.yA.x(x e. y <-> (x e. z /\ ph)) <-> E.yA.x(x e. y <-> (x e. w /\ (x e. z /\ ph)))))
12	2, 11	mpbiri 211	. . 3 |- (w = z -> E.yA.x(x e. y <-> (x e. z /\ ph)))
13	12	19.23aiv 1674	. 2 |- (E.w w = z -> E.yA.x(x e. y <-> (x e. z /\ ph)))
14	1, 13	ax-mp 7	1 |- E.yA.x(x e. y <-> (x e. z /\ ph))

Syntax hints:   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-8 1306  ax-9 1307  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-sep 3438

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327

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