Theorem axrep 1473

Description: Axiom of Replacement expressed more compactly, with fewest number of different variables.

Assertion

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

Distinct variable group(s):   x,y,z

Proof of Theorem axrep

Step	Hyp	Ref	Expression
1		visset 1350	. . 3 |- y e. V
2		eleq2 1150	. . . . . . . . 9 |- (w = y -> (x e. w <-> x e. y))
3	2	anbi1d 469	. . . . . . . 8 |- (w = y -> ((x e. w /\ A.yph) <-> (x e. y /\ A.yph)))
4	3	biexdv 936	. . . . . . 7 |- (w = y -> (E.x(x e. w /\ A.yph) <-> E.x(x e. y /\ A.yph)))
5	4	bibi2d 470	. . . . . 6 |- (w = y -> ((z e. x <-> E.x(x e. w /\ A.yph)) <-> (z e. x <-> E.x(x e. y /\ A.yph))))
6	5	bialdv 935	. . . . 5 |- (w = y -> (A.z(z e. x <-> E.x(x e. w /\ A.yph)) <-> A.z(z e. x <-> E.x(x e. y /\ A.yph))))
7	6	biexdv 936	. . . 4 |- (w = y -> (E.xA.z(z e. x <-> E.x(x e. w /\ A.yph)) <-> E.xA.z(z e. x <-> E.x(x e. y /\ A.yph))))
8	7	imbi2d 464	. . 3 |- (w = y -> ((A.xE.yA.z(ph -> z = y) -> E.xA.z(z e. x <-> E.x(x e. w /\ A.yph))) <-> (A.xE.yA.z(ph -> z = y) -> E.xA.z(z e. x <-> E.x(x e. y /\ A.yph)))))
9		ax-4 673	. . . . . . . . 9 |- (A.yph -> ph)
10	9	syl4 19	. . . . . . . 8 |- ((ph -> z = y) -> (A.yph -> z = y))
11	10	19.20i 691	. . . . . . 7 |- (A.z(ph -> z = y) -> A.z(A.yph -> z = y))
12	11	19.22i 723	. . . . . 6 |- (E.yA.z(ph -> z = y) -> E.yA.z(A.yph -> z = y))
13	12	19.20i 691	. . . . 5 |- (A.xE.yA.z(ph -> z = y) -> A.xE.yA.z(A.yph -> z = y))
14		ax-rep 1075	. . . . 5 |- (A.xE.yA.z(A.yph -> z = y) -> E.yA.z(z e. y <-> E.x(x e. w /\ A.yph)))
15	13, 14	syl 12	. . . 4 |- (A.xE.yA.z(ph -> z = y) -> E.yA.z(z e. y <-> E.x(x e. w /\ A.yph)))
16		ax-17 925	. . . . . . 7 |- (z e. y -> A.x z e. y)
17		hbe1 709	. . . . . . 7 |- (E.x(x e. w /\ A.yph) -> A.xE.x(x e. w /\ A.yph))
18	16, 17	hbbi 705	. . . . . 6 |- ((z e. y <-> E.x(x e. w /\ A.yph)) -> A.x(z e. y <-> E.x(x e. w /\ A.yph)))
19	18	hbal 700	. . . . 5 |- (A.z(z e. y <-> E.x(x e. w /\ A.yph)) -> A.xA.z(z e. y <-> E.x(x e. w /\ A.yph)))
20		ax-17 925	. . . . . . 7 |- (z e. x -> A.y z e. x)
21		ax-17 925	. . . . . . . . 9 |- (x e. w -> A.y x e. w)
22		hba1 698	. . . . . . . . 9 |- (A.yph -> A.yA.yph)
23	21, 22	hban 704	. . . . . . . 8 |- ((x e. w /\ A.yph) -> A.y(x e. w /\ A.yph))
24	23	hbex 701	. . . . . . 7 |- (E.x(x e. w /\ A.yph) -> A.yE.x(x e. w /\ A.yph))
25	20, 24	hbbi 705	. . . . . 6 |- ((z e. x <-> E.x(x e. w /\ A.yph)) -> A.y(z e. x <-> E.x(x e. w /\ A.yph)))
26	25	hbal 700	. . . . 5 |- (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)))
27		eleq2 1150	. . . . . . 7 |- (y = x -> (z e. y <-> z e. x))
28	27	bibi1d 471	. . . . . 6 |- (y = x -> ((z e. y <-> E.x(x e. w /\ A.yph)) <-> (z e. x <-> E.x(x e. w /\ A.yph))))
29	28	bialdv 935	. . . . 5 |- (y = x -> (A.z(z e. y <-> E.x(x e. w /\ A.yph)) <-> A.z(z e. x <-> E.x(x e. w /\ A.yph))))
30	19, 26, 29	cbvex 849	. . . 4 |- (E.yA.z(z e. y <-> E.x(x e. w /\ A.yph)) <-> E.xA.z(z e. x <-> E.x(x e. w /\ A.yph)))
31	15, 30	sylib 173	. . 3 |- (A.xE.yA.z(ph -> z = y) -> E.xA.z(z e. x <-> E.x(x e. w /\ A.yph)))
32	1, 8, 31	vtocl 1378	. 2 |- (A.xE.yA.z(ph -> z = y) -> E.xA.z(z e. x <-> E.x(x e. y /\ A.yph)))
33	32	19.35ri 756	1 |- E.x(E.yA.z(ph -> z = y) -> A.z(z e. x <-> E.x(x e. y /\ 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:  axrep2 1474  axrepndlem1 3738

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-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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