Theorem axsep2 2699

Description: A less restrictive version of the Separation Scheme axsep 2697, 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 2698 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 1123	. 2 |- E.w w = z
2		ax-sep 2698	. . . 4 |- E.yA.x(x e. y <-> (x e. w /\ (x e. z /\ ph)))
3		elequ2 1135	. . . . . . . . . . 11 |- (w = z -> (x e. w <-> x e. z))
4	3	biimprd 154	. . . . . . . . . 10 |- (w = z -> (x e. z -> x e. w))
5	4	pm4.71rd 638	. . . . . . . . 9 |- (w = z -> (x e. z <-> (x e. w /\ x e. z)))
6	5	anbi1d 616	. . . . . . . 8 |- (w = z -> ((x e. z /\ ph) <-> ((x e. w /\ x e. z) /\ ph)))
7		anass 439	. . . . . . . 8 |- (((x e. w /\ x e. z) /\ ph) <-> (x e. w /\ (x e. z /\ ph)))
8	6, 7	syl6bb 535	. . . . . . 7 |- (w = z -> ((x e. z /\ ph) <-> (x e. w /\ (x e. z /\ ph))))
9	8	bibi2d 617	. . . . . 6 |- (w = z -> ((x e. y <-> (x e. z /\ ph)) <-> (x e. y <-> (x e. w /\ (x e. z /\ ph)))))
10	9	albidv 1276	. . . . 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 1277	. . . 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 194	. . 3 |- (w = z -> E.yA.x(x e. y <-> (x e. z /\ ph)))
13	12	19.23aiv 1293	. 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 146   /\ wa 223  A.wal 952   = wceq 954   e. wcel 956  E.wex 978

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 961  ax-8 962  ax-9 963  ax-12 966  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-sep 2698

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979

Copyright terms: Public domain