Theorem axrepnd 6098

Description: A version of the Axiom of Replacement with no distinct variable conditions.

Assertion

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

Proof of Theorem axrepnd

Step	Hyp	Ref	Expression
1		axrepndlem2 6097	. . . 4 |- (((-. A.x x = y /\ -. A.x x = z) /\ -. A.y y = z) -> E.x(E.yA.z(ph -> z = y) -> A.z(z e. x <-> E.x(x e. y /\ A.yph))))
2		hbnae 1507	. . . . . . 7 |- (-. A.x x = y -> A.x -. A.x x = y)
3		hbnae 1507	. . . . . . 7 |- (-. A.x x = z -> A.x -. A.x x = z)
4	2, 3	hban 1356	. . . . . 6 |- ((-. A.x x = y /\ -. A.x x = z) -> A.x(-. A.x x = y /\ -. A.x x = z))
5		hbnae 1507	. . . . . 6 |- (-. A.y y = z -> A.x -. A.y y = z)
6	4, 5	hban 1356	. . . . 5 |- (((-. A.x x = y /\ -. A.x x = z) /\ -. A.y y = z) -> A.x((-. A.x x = y /\ -. A.x x = z) /\ -. A.y y = z))
7		hbnae 1507	. . . . . . . . 9 |- (-. A.x x = y -> A.z -. A.x x = y)
8		hbnae 1507	. . . . . . . . 9 |- (-. A.x x = z -> A.z -. A.x x = z)
9	7, 8	hban 1356	. . . . . . . 8 |- ((-. A.x x = y /\ -. A.x x = z) -> A.z(-. A.x x = y /\ -. A.x x = z))
10		hbnae 1507	. . . . . . . 8 |- (-. A.y y = z -> A.z -. A.y y = z)
11	9, 10	hban 1356	. . . . . . 7 |- (((-. A.x x = y /\ -. A.x x = z) /\ -. A.y y = z) -> A.z((-. A.x x = y /\ -. A.x x = z) /\ -. A.y y = z))
12		ax-15 1751	. . . . . . . . . . . . 13 |- (-. A.y y = z -> (-. A.y y = x -> (z e. x -> A.y z e. x)))
13	12	com12 14	. . . . . . . . . . . 12 |- (-. A.y y = x -> (-. A.y y = z -> (z e. x -> A.y z e. x)))
14	13	nalequcoms 1504	. . . . . . . . . . 11 |- (-. A.x x = y -> (-. A.y y = z -> (z e. x -> A.y z e. x)))
15	14	imp 377	. . . . . . . . . 10 |- ((-. A.x x = y /\ -. A.y y = z) -> (z e. x -> A.y z e. x))
16	15	adantlr 429	. . . . . . . . 9 |- (((-. A.x x = y /\ -. A.x x = z) /\ -. A.y y = z) -> (z e. x -> A.y z e. x))
17		ax-4 1319	. . . . . . . . 9 |- (A.y z e. x -> z e. x)
18	16, 17	impbid1 575	. . . . . . . 8 |- (((-. A.x x = y /\ -. A.x x = z) /\ -. A.y y = z) -> (z e. x <-> A.y z e. x))
19		ax-15 1751	. . . . . . . . . . . . . 14 |- (-. A.z z = x -> (-. A.z z = y -> (x e. y -> A.z x e. y)))
20	19	imp 377	. . . . . . . . . . . . 13 |- ((-. A.z z = x /\ -. A.z z = y) -> (x e. y -> A.z x e. y))
21		alequcom 1502	. . . . . . . . . . . . . 14 |- (A.z z = x -> A.x x = z)
22	21	con3i 114	. . . . . . . . . . . . 13 |- (-. A.x x = z -> -. A.z z = x)
23		alequcom 1502	. . . . . . . . . . . . . 14 |- (A.z z = y -> A.y y = z)
24	23	con3i 114	. . . . . . . . . . . . 13 |- (-. A.y y = z -> -. A.z z = y)
25	20, 22, 24	syl2an 503	. . . . . . . . . . . 12 |- ((-. A.x x = z /\ -. A.y y = z) -> (x e. y -> A.z x e. y))
26	25	adantll 428	. . . . . . . . . . 11 |- (((-. A.x x = y /\ -. A.x x = z) /\ -. A.y y = z) -> (x e. y -> A.z x e. y))
27		ax-4 1319	. . . . . . . . . . 11 |- (A.z x e. y -> x e. y)
28	26, 27	impbid1 575	. . . . . . . . . 10 |- (((-. A.x x = y /\ -. A.x x = z) /\ -. A.y y = z) -> (x e. y <-> A.z x e. y))
29	28	anbi1d 679	. . . . . . . . 9 |- (((-. A.x x = y /\ -. A.x x = z) /\ -. A.y y = z) -> ((x e. y /\ A.yph) <-> (A.z x e. y /\ A.yph)))
30	6, 29	exbid 1460	. . . . . . . 8 |- (((-. A.x x = y /\ -. A.x x = z) /\ -. A.y y = z) -> (E.x(x e. y /\ A.yph) <-> E.x(A.z x e. y /\ A.yph)))
31	18, 30	bibi12d 691	. . . . . . 7 |- (((-. A.x x = y /\ -. A.x x = z) /\ -. A.y y = z) -> ((z e. x <-> E.x(x e. y /\ A.yph)) <-> (A.y z e. x <-> E.x(A.z x e. y /\ A.yph))))
32	11, 31	albid 1459	. . . . . 6 |- (((-. A.x x = y /\ -. A.x x = z) /\ -. A.y y = z) -> (A.z(z e. x <-> E.x(x e. y /\ A.yph)) <-> A.z(A.y z e. x <-> E.x(A.z x e. y /\ A.yph))))
33	32	imbi2d 674	. . . . 5 |- (((-. A.x x = y /\ -. A.x x = z) /\ -. A.y y = z) -> ((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(A.y z e. x <-> E.x(A.z x e. y /\ A.yph)))))
34	6, 33	exbid 1460	. . . 4 |- (((-. A.x x = y /\ -. A.x x = z) /\ -. A.y y = z) -> (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(A.y z e. x <-> E.x(A.z x e. y /\ A.yph)))))
35	1, 34	mpbid 212	. . 3 |- (((-. A.x x = y /\ -. A.x x = z) /\ -. A.y y = z) -> E.x(E.yA.z(ph -> z = y) -> A.z(A.y z e. x <-> E.x(A.z x e. y /\ A.yph))))
36	35	exp31 407	. 2 |- (-. A.x x = y -> (-. A.x x = z -> (-. A.y y = z -> E.x(E.yA.z(ph -> z = y) -> A.z(A.y z e. x <-> E.x(A.z x e. y /\ A.yph))))))
37		hbae 1505	. . . . 5 |- (A.x x = y -> A.zA.x x = y)
38		nd2 6091	. . . . . . 7 |- (A.y y = x -> -. A.y z e. x)
39	38	alequcoms 1503	. . . . . 6 |- (A.x x = y -> -. A.y z e. x)
40		hbae 1505	. . . . . . 7 |- (A.x x = y -> A.xA.x x = y)
41		nd3 6092	. . . . . . . 8 |- (A.x x = y -> -. A.z x e. y)
42	41	intnanrd 758	. . . . . . 7 |- (A.x x = y -> -. (A.z x e. y /\ A.yph))
43	40, 42	nexd 1457	. . . . . 6 |- (A.x x = y -> -. E.x(A.z x e. y /\ A.yph))
44		pm5.21 741	. . . . . 6 |- ((-. A.y z e. x /\ -. E.x(A.z x e. y /\ A.yph)) -> (A.y z e. x <-> E.x(A.z x e. y /\ A.yph)))
45	39, 43, 44	syl11anc 524	. . . . 5 |- (A.x x = y -> (A.y z e. x <-> E.x(A.z x e. y /\ A.yph)))
46	37, 45	19.21ai 1345	. . . 4 |- (A.x x = y -> A.z(A.y z e. x <-> E.x(A.z x e. y /\ A.yph)))
47	46	a1d 15	. . 3 |- (A.x x = y -> (E.yA.z(ph -> z = y) -> A.z(A.y z e. x <-> E.x(A.z x e. y /\ A.yph))))
48		19.8a 1376	. . 3 |- ((E.yA.z(ph -> z = y) -> A.z(A.y z e. x <-> E.x(A.z x e. y /\ A.yph))) -> E.x(E.yA.z(ph -> z = y) -> A.z(A.y z e. x <-> E.x(A.z x e. y /\ A.yph))))
49	47, 48	syl 12	. 2 |- (A.x x = y -> E.x(E.yA.z(ph -> z = y) -> A.z(A.y z e. x <-> E.x(A.z x e. y /\ A.yph))))
50		hbae 1505	. . . . 5 |- (A.x x = z -> A.zA.x x = z)
51		nd4 6093	. . . . . 6 |- (A.x x = z -> -. A.y z e. x)
52		hbae 1505	. . . . . . 7 |- (A.x x = z -> A.xA.x x = z)
53		nd1 6090	. . . . . . . . 9 |- (A.z z = x -> -. A.z x e. y)
54	53	alequcoms 1503	. . . . . . . 8 |- (A.x x = z -> -. A.z x e. y)
55	54	intnanrd 758	. . . . . . 7 |- (A.x x = z -> -. (A.z x e. y /\ A.yph))
56	52, 55	nexd 1457	. . . . . 6 |- (A.x x = z -> -. E.x(A.z x e. y /\ A.yph))
57	51, 56, 44	syl11anc 524	. . . . 5 |- (A.x x = z -> (A.y z e. x <-> E.x(A.z x e. y /\ A.yph)))
58	50, 57	19.21ai 1345	. . . 4 |- (A.x x = z -> A.z(A.y z e. x <-> E.x(A.z x e. y /\ A.yph)))
59	58	a1d 15	. . 3 |- (A.x x = z -> (E.yA.z(ph -> z = y) -> A.z(A.y z e. x <-> E.x(A.z x e. y /\ A.yph))))
60	59, 48	syl 12	. 2 |- (A.x x = z -> E.x(E.yA.z(ph -> z = y) -> A.z(A.y z e. x <-> E.x(A.z x e. y /\ A.yph))))
61		hbae 1505	. . . . 5 |- (A.y y = z -> A.zA.y y = z)
62		nd1 6090	. . . . . 6 |- (A.y y = z -> -. A.y z e. x)
63		hbae 1505	. . . . . . 7 |- (A.y y = z -> A.xA.y y = z)
64		nd2 6091	. . . . . . . . 9 |- (A.z z = y -> -. A.z x e. y)
65	64	alequcoms 1503	. . . . . . . 8 |- (A.y y = z -> -. A.z x e. y)
66	65	intnanrd 758	. . . . . . 7 |- (A.y y = z -> -. (A.z x e. y /\ A.yph))
67	63, 66	nexd 1457	. . . . . 6 |- (A.y y = z -> -. E.x(A.z x e. y /\ A.yph))
68	62, 67, 44	syl11anc 524	. . . . 5 |- (A.y y = z -> (A.y z e. x <-> E.x(A.z x e. y /\ A.yph)))
69	61, 68	19.21ai 1345	. . . 4 |- (A.y y = z -> A.z(A.y z e. x <-> E.x(A.z x e. y /\ A.yph)))
70	69	a1d 15	. . 3 |- (A.y y = z -> (E.yA.z(ph -> z = y) -> A.z(A.y z e. x <-> E.x(A.z x e. y /\ A.yph))))
71	70, 48	syl 12	. 2 |- (A.y y = z -> E.x(E.yA.z(ph -> z = y) -> A.z(A.y z e. x <-> E.x(A.z x e. y /\ A.yph))))
72	36, 49, 60, 71	pm2.61iii 147	1 |- E.x(E.yA.z(ph -> z = y) -> A.z(A.y z e. x <-> E.x(A.z x e. y /\ A.yph)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326

This theorem is referenced by:  zfcndrep 6118  axrepprim 13786

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-15 1751  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-reg 5695

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049

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