Theorem ralxp 4041

Description: Universal quantification restricted to a cross product is equivalent to a double restricted quantification. The hypothesis specifies an implicit substitution.

Hypothesis

Ref	Expression
ralxp.1	|- (x = <.y, z>. -> (ph <-> ps))

Assertion

Ref	Expression
ralxp	|- (A.x e. (A X. B)ph <-> A.y e. A A.z e. B ps)

Distinct variable groups:   x,y,z,A   x,B,y,z   ph,y,z   ps,x

Proof of Theorem ralxp

Step	Hyp	Ref	Expression
1		ralxp.1	. . . . 5 |- (x = <.y, z>. -> (ph <-> ps))
2	1	rcla4cv 2377	. . . 4 |- (A.x e. (A X. B)ph -> (<.y, z>. e. (A X. B) -> ps))
3		visset 2295	. . . . 5 |- z e. _V
4	3	opelxp 4036	. . . 4 |- (<.y, z>. e. (A X. B) <-> (y e. A /\ z e. B))
5	2, 4	syl5ibr 224	. . 3 |- (A.x e. (A X. B)ph -> ((y e. A /\ z e. B) -> ps))
6	5	r19.21aivv 2183	. 2 |- (A.x e. (A X. B)ph -> A.y e. A A.z e. B ps)
7		elxp 4018	. . . . . 6 |- (x e. (A X. B) <-> E.yE.z(x = <.y, z>. /\ (y e. A /\ z e. B)))
8		simpl 346	. . . . . . 7 |- ((x = <.y, z>. /\ (y e. A /\ z e. B)) -> x = <.y, z>.)
9	8	2eximi 1388	. . . . . 6 |- (E.yE.z(x = <.y, z>. /\ (y e. A /\ z e. B)) -> E.yE.z x = <.y, z>.)
10	7, 9	sylbi 216	. . . . 5 |- (x e. (A X. B) -> E.yE.z x = <.y, z>.)
11		hbra1 2147	. . . . . . 7 |- (A.y e. A A.z e. B ps -> A.yA.y e. A A.z e. B ps)
12		ax-17 1317	. . . . . . 7 |- ((x e. (A X. B) -> ph) -> A.y(x e. (A X. B) -> ph))
13	11, 12	hbim 1354	. . . . . 6 |- ((A.y e. A A.z e. B ps -> (x e. (A X. B) -> ph)) -> A.y(A.y e. A A.z e. B ps -> (x e. (A X. B) -> ph)))
14		ax-17 1317	. . . . . . . . 9 |- (y e. A -> A.z y e. A)
15		hbra1 2147	. . . . . . . . 9 |- (A.z e. B ps -> A.zA.z e. B ps)
16	14, 15	hbral 2146	. . . . . . . 8 |- (A.y e. A A.z e. B ps -> A.zA.y e. A A.z e. B ps)
17		ax-17 1317	. . . . . . . 8 |- ((x e. (A X. B) -> ph) -> A.z(x e. (A X. B) -> ph))
18	16, 17	hbim 1354	. . . . . . 7 |- ((A.y e. A A.z e. B ps -> (x e. (A X. B) -> ph)) -> A.z(A.y e. A A.z e. B ps -> (x e. (A X. B) -> ph)))
19		eleq1 1957	. . . . . . . . . 10 |- (x = <.y, z>. -> (x e. (A X. B) <-> <.y, z>. e. (A X. B)))
20	19, 4	syl6bb 595	. . . . . . . . 9 |- (x = <.y, z>. -> (x e. (A X. B) <-> (y e. A /\ z e. B)))
21	20, 1	imbi12d 688	. . . . . . . 8 |- (x = <.y, z>. -> ((x e. (A X. B) -> ph) <-> ((y e. A /\ z e. B) -> ps)))
22		ra42 2157	. . . . . . . 8 |- (A.y e. A A.z e. B ps -> ((y e. A /\ z e. B) -> ps))
23	21, 22	syl5bir 227	. . . . . . 7 |- (x = <.y, z>. -> (A.y e. A A.z e. B ps -> (x e. (A X. B) -> ph)))
24	18, 23	19.23ai 1412	. . . . . 6 |- (E.z x = <.y, z>. -> (A.y e. A A.z e. B ps -> (x e. (A X. B) -> ph)))
25	13, 24	19.23ai 1412	. . . . 5 |- (E.yE.z x = <.y, z>. -> (A.y e. A A.z e. B ps -> (x e. (A X. B) -> ph)))
26	10, 25	syl 12	. . . 4 |- (x e. (A X. B) -> (A.y e. A A.z e. B ps -> (x e. (A X. B) -> ph)))
27	26	pm2.43b 81	. . 3 |- (A.y e. A A.z e. B ps -> (x e. (A X. B) -> ph))
28	27	r19.21aiv 2175	. 2 |- (A.y e. A A.z e. B ps -> A.x e. (A X. B)ph)
29	6, 28	impbii 174	1 |- (A.x e. (A X. B)ph <-> A.y e. A A.z e. B ps)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  A.wral 2105  <.cop 3046   X. cxp 3984

This theorem is referenced by:  rexxp 4042  ralxpf 4043  ffnoprv 4943  eqfnoprv 4945  f1stres 5034  f2ndres 5035  rankxplim 5823  ref4w 14370  tarcrpr 15237  f1opr 15714  cnresoprab 15915  reparphtlem2 16064

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-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-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-opab 3396  df-xp 4000

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