Theorem axsep2 2778

Description: A less restrictive version of the Separation Scheme axsep 2776, 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 2777 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 1161	. 2 |- E.w w = z
2		ax-sep 2777	. . . 4 |- E.yA.x(x e. y <-> (x e. w /\ (x e. z /\ ph)))
3		elequ2 1174	. . . . . . . . . . 11 |- (w = z -> (x e. w <-> x e. z))
4	3	biimprd 152	. . . . . . . . . 10 |- (w = z -> (x e. z -> x e. w))
5	4	pm4.71rd 642	. . . . . . . . 9 |- (w = z -> (x e. z <-> (x e. w /\ x e. z)))
6	5	anbi1d 620	. . . . . . . 8 |- (w = z -> ((x e. z /\ ph) <-> ((x e. w /\ x e. z) /\ ph)))
7		anass 441	. . . . . . . 8 |- (((x e. w /\ x e. z) /\ ph) <-> (x e. w /\ (x e. z /\ ph)))
8	6, 7	syl6bb 539	. . . . . . 7 |- (w = z -> ((x e. z /\ ph) <-> (x e. w /\ (x e. z /\ ph))))
9	8	bibi2d 621	. . . . . 6 |- (w = z -> ((x e. y <-> (x e. z /\ ph)) <-> (x e. y <-> (x e. w /\ (x e. z /\ ph)))))
10	9	albidv 1316	. . . . 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 1317	. . . 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 192	. . 3 |- (w = z -> E.yA.x(x e. y <-> (x e. z /\ ph)))
13	12	19.23aiv 1333	. 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 144   /\ wa 221  A.wal 990   = wceq 992   e. wcel 994  E.wex 1016

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 999  ax-8 1000  ax-9 1001  ax-12 1004  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-sep 2777

This theorem depends on definitions:  df-bi 145  df-an 223  df-ex 1017

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