Theorem axrep3 2691

Description: Axiom of Replacement slightly strengthened from axrep2 2690; w may occur free in ph.

Assertion

Ref	Expression
axrep3	|- E.x(E.yA.z(ph -> z = y) -> A.z(z e. x <-> E.x(x e. w /\ A.yph)))

Distinct variable group:   x,w,y,z

Proof of Theorem axrep3

Step	Hyp	Ref	Expression
1		hbe1 1014	. . . 4 |- (E.yA.z(ph -> z = y) -> A.yE.yA.z(ph -> z = y))
2		ax-17 969	. . . . . 6 |- (z e. x -> A.y z e. x)
3		ax-17 969	. . . . . . . 8 |- (x e. w -> A.y x e. w)
4		hba1 1001	. . . . . . . 8 |- (A.yph -> A.yA.yph)
5	3, 4	hban 1007	. . . . . . 7 |- ((x e. w /\ A.yph) -> A.y(x e. w /\ A.yph))
6	5	hbex 1004	. . . . . 6 |- (E.x(x e. w /\ A.yph) -> A.yE.x(x e. w /\ A.yph))
7	2, 6	hbbi 1008	. . . . 5 |- ((z e. x <-> E.x(x e. w /\ A.yph)) -> A.y(z e. x <-> E.x(x e. w /\ A.yph)))
8	7	hbal 1003	. . . 4 |- (A.z(z e. x <-> E.x(x e. w /\ A.yph)) -> A.yA.z(z e. x <-> E.x(x e. w /\ A.yph)))
9	1, 8	hbim 1005	. . 3 |- ((E.yA.z(ph -> z = y) -> A.z(z e. x <-> E.x(x e. w /\ A.yph))) -> A.y(E.yA.z(ph -> z = y) -> A.z(z e. x <-> E.x(x e. w /\ A.yph))))
10	9	hbex 1004	. 2 |- (E.x(E.yA.z(ph -> z = y) -> A.z(z e. x <-> E.x(x e. w /\ A.yph))) -> A.yE.x(E.yA.z(ph -> z = y) -> A.z(z e. x <-> E.x(x e. w /\ A.yph))))
11		elequ2 1135	. . . . . . . 8 |- (y = w -> (x e. y <-> x e. w))
12	11	anbi1d 616	. . . . . . 7 |- (y = w -> ((x e. y /\ A.yph) <-> (x e. w /\ A.yph)))
13	12	exbidv 1277	. . . . . 6 |- (y = w -> (E.x(x e. y /\ A.yph) <-> E.x(x e. w /\ A.yph)))
14	13	bibi2d 617	. . . . 5 |- (y = w -> ((z e. x <-> E.x(x e. y /\ A.yph)) <-> (z e. x <-> E.x(x e. w /\ A.yph))))
15	14	albidv 1276	. . . 4 |- (y = w -> (A.z(z e. x <-> E.x(x e. y /\ A.yph)) <-> A.z(z e. x <-> E.x(x e. w /\ A.yph))))
16	15	imbi2d 611	. . 3 |- (y = w -> ((E.yA.z(ph -> z = y) -> A.z(z e. x <-> E.x(x e. y /\ A.yph))) <-> (E.yA.z(ph -> z = y) -> A.z(z e. x <-> E.x(x e. w /\ A.yph)))))
17	16	exbidv 1277	. 2 |- (y = w -> (E.x(E.yA.z(ph -> z = y) -> A.z(z e. x <-> E.x(x e. y /\ A.yph))) <-> E.x(E.yA.z(ph -> z = y) -> A.z(z e. x <-> E.x(x e. w /\ A.yph)))))
18		axrep2 2690	. 2 |- E.x(E.yA.z(ph -> z = y) -> A.z(z e. x <-> E.x(x e. y /\ A.yph)))
19	10, 17, 18	chvar 1165	1 |- E.x(E.yA.z(ph -> z = y) -> A.z(z e. x <-> E.x(x e. w /\ A.yph)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 952   = wceq 954   e. wcel 956  E.wex 978

This theorem is referenced by:  axrep4 2692

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-12 966  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-rep 2688

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

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