Theorem axrep2 1474

Description: Axiom of Replacement slightly strengthened from axrep 1473; w may occur free in ph.

Assertion

Ref	Expression
axrep2	|- E.x(E.yA.z(ph -> z = y) -> A.z(z e. x <-> E.x(x e. w /\ A.yph)))

Distinct variable group(s):   x,y,z,w

Proof of Theorem axrep2

Step	Hyp	Ref	Expression
1		hbe1 709	. . . 4 |- (E.yA.z(ph -> z = y) -> A.yE.yA.z(ph -> z = y))
2		ax-17 925	. . . . . 6 |- (z e. x -> A.y z e. x)
3		ax-17 925	. . . . . . . 8 |- (x e. w -> A.y x e. w)
4		hba1 698	. . . . . . . 8 |- (A.yph -> A.yA.yph)
5	3, 4	hban 704	. . . . . . 7 |- ((x e. w /\ A.yph) -> A.y(x e. w /\ A.yph))
6	5	hbex 701	. . . . . 6 |- (E.x(x e. w /\ A.yph) -> A.yE.x(x e. w /\ A.yph))
7	2, 6	hbbi 705	. . . . 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 700	. . . 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 702	. . 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 701	. 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		visset 1350	. 2 |- w e. V
12		a14b 820	. . . . . . . 8 |- (y = w -> (x e. y <-> x e. w))
13	12	anbi1d 469	. . . . . . 7 |- (y = w -> ((x e. y /\ A.yph) <-> (x e. w /\ A.yph)))
14	13	biexdv 936	. . . . . 6 |- (y = w -> (E.x(x e. y /\ A.yph) <-> E.x(x e. w /\ A.yph)))
15	14	bibi2d 470	. . . . 5 |- (y = w -> ((z e. x <-> E.x(x e. y /\ A.yph)) <-> (z e. x <-> E.x(x e. w /\ A.yph))))
16	15	bialdv 935	. . . 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))))
17	16	imbi2d 464	. . 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)))))
18	17	biexdv 936	. 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)))))
19		axrep 1473	. 2 |- E.x(E.yA.z(ph -> z = y) -> A.z(z e. x <-> E.x(x e. y /\ A.yph)))
20	10, 11, 18, 19	vtoclf 1377	1 |- E.x(E.yA.z(ph -> z = y) -> A.z(z e. x <-> E.x(x e. w /\ A.yph)))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   = weq 797   e. wel 803

This theorem is referenced by:  zfrep2 1475

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-12 802  ax-14 805  ax-17 925  ax-ext 1074  ax-rep 1075

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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