Theorem zfrep2 1475

Description: A more traditional version of the Axiom of Replacement.

Hypothesis

Ref	Expression
zfrep2.1	|- (ph -> A.zph)

Assertion

Ref	Expression
zfrep2	|- (A.xE.zA.y(ph -> y = z) -> E.zA.y(y e. z <-> E.x(x e. w /\ ph)))

Distinct variable group(s):   x,y,z,w

Proof of Theorem zfrep2

Step	Hyp	Ref	Expression
1		axrep2 1474	. . 3 |- E.x(E.zA.y(ph -> y = z) -> A.y(y e. x <-> E.x(x e. w /\ A.zph)))
2	1	19.35i 755	. 2 |- (A.xE.zA.y(ph -> y = z) -> E.xA.y(y e. x <-> E.x(x e. w /\ A.zph)))
3		ax-17 925	. . . . 5 |- (y e. x -> A.z y e. x)
4		ax-17 925	. . . . . . 7 |- (x e. w -> A.z x e. w)
5		hba1 698	. . . . . . 7 |- (A.zph -> A.zA.zph)
6	4, 5	hban 704	. . . . . 6 |- ((x e. w /\ A.zph) -> A.z(x e. w /\ A.zph))
7	6	hbex 701	. . . . 5 |- (E.x(x e. w /\ A.zph) -> A.zE.x(x e. w /\ A.zph))
8	3, 7	hbbi 705	. . . 4 |- ((y e. x <-> E.x(x e. w /\ A.zph)) -> A.z(y e. x <-> E.x(x e. w /\ A.zph)))
9	8	hbal 700	. . 3 |- (A.y(y e. x <-> E.x(x e. w /\ A.zph)) -> A.zA.y(y e. x <-> E.x(x e. w /\ A.zph)))
10		ax-17 925	. . . . 5 |- (y e. z -> A.x y e. z)
11		hbe1 709	. . . . 5 |- (E.x(x e. w /\ ph) -> A.xE.x(x e. w /\ ph))
12	10, 11	hbbi 705	. . . 4 |- ((y e. z <-> E.x(x e. w /\ ph)) -> A.x(y e. z <-> E.x(x e. w /\ ph)))
13	12	hbal 700	. . 3 |- (A.y(y e. z <-> E.x(x e. w /\ ph)) -> A.xA.y(y e. z <-> E.x(x e. w /\ ph)))
14		ax-17 925	. . . 4 |- (x = z -> A.y x = z)
15		a14b 820	. . . . 5 |- (x = z -> (y e. x <-> y e. z))
16		ax-4 673	. . . . . . . . 9 |- (A.zph -> ph)
17		zfrep2.1	. . . . . . . . 9 |- (ph -> A.zph)
18	16, 17	impbi 139	. . . . . . . 8 |- (A.zph <-> ph)
19	18	anbi2i 367	. . . . . . 7 |- ((x e. w /\ A.zph) <-> (x e. w /\ ph))
20	19	biex 733	. . . . . 6 |- (E.x(x e. w /\ A.zph) <-> E.x(x e. w /\ ph))
21	20	a1i 7	. . . . 5 |- (x = z -> (E.x(x e. w /\ A.zph) <-> E.x(x e. w /\ ph)))
22	15, 21	bibi12d 477	. . . 4 |- (x = z -> ((y e. x <-> E.x(x e. w /\ A.zph)) <-> (y e. z <-> E.x(x e. w /\ ph))))
23	14, 22	biald 782	. . 3 |- (x = z -> (A.y(y e. x <-> E.x(x e. w /\ A.zph)) <-> A.y(y e. z <-> E.x(x e. w /\ ph))))
24	9, 13, 23	cbvex 849	. 2 |- (E.xA.y(y e. x <-> E.x(x e. w /\ A.zph)) <-> E.zA.y(y e. z <-> E.x(x e. w /\ ph)))
25	2, 24	sylib 173	1 |- (A.xE.zA.y(ph -> y = z) -> E.zA.y(y e. z <-> E.x(x e. w /\ ph)))

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

This theorem is referenced by:  zfrep3 1476  funimaexg 2715

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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