Theorem ralxp 3218

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 1874	. . . 4 |- (A.x e. (A X. B)ph -> (<.y, z>. e. (A X. B) -> ps))
3		visset 1813	. . . . 5 |- z e. V
4	3	opelxp 3214	. . . 4 |- (<.y, z>. e. (A X. B) <-> (y e. A /\ z e. B))
5	2, 4	syl5ibr 207	. . 3 |- (A.x e. (A X. B)ph -> ((y e. A /\ z e. B) -> ps))
6	5	r19.21aivv 1720	. 2 |- (A.x e. (A X. B)ph -> A.y e. A A.z e. B ps)
7		elxp 3202	. . . . . 6 |- (x e. (A X. B) <-> E.yE.z(x = <.y, z>. /\ (y e. A /\ z e. B)))
8		pm3.26 319	. . . . . . 7 |- ((x = <.y, z>. /\ (y e. A /\ z e. B)) -> x = <.y, z>.)
9	8	19.22i2 1041	. . . . . 6 |- (E.yE.z(x = <.y, z>. /\ (y e. A /\ z e. B)) -> E.yE.z x = <.y, z>.)
10	7, 9	sylbi 199	. . . . 5 |- (x e. (A X. B) -> E.yE.z x = <.y, z>.)
11		hbra1 1687	. . . . . . 7 |- (A.y e. A A.z e. B ps -> A.yA.y e. A A.z e. B ps)
12		ax-17 971	. . . . . . 7 |- ((x e. (A X. B) -> ph) -> A.y(x e. (A X. B) -> ph))
13	11, 12	hbim 1007	. . . . . 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-17 971	. . . . . . . . 9 |- (y e. A -> A.z y e. A)
15		hbra1 1687	. . . . . . . . 9 |- (A.z e. B ps -> A.zA.z e. B ps)
16	14, 15	hbral 1686	. . . . . . . 8 |- (A.y e. A A.z e. B ps -> A.zA.y e. A A.z e. B ps)
17		ax-17 971	. . . . . . . 8 |- ((x e. (A X. B) -> ph) -> A.z(x e. (A X. B) -> ph))
18	16, 17	hbim 1007	. . . . . . 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 1534	. . . . . . . . . 10 |- (x = <.y, z>. -> (x e. (A X. B) <-> <.y, z>. e. (A X. B)))
20	19, 4	syl6bb 536	. . . . . . . . 9 |- (x = <.y, z>. -> (x e. (A X. B) <-> (y e. A /\ z e. B)))
21	20, 1	imbi12d 626	. . . . . . . 8 |- (x = <.y, z>. -> ((x e. (A X. B) -> ph) <-> ((y e. A /\ z e. B) -> ps)))
22		ra42 1696	. . . . . . . 8 |- (A.y e. A A.z e. B ps -> ((y e. A /\ z e. B) -> ps))
23	21, 22	syl5bir 210	. . . . . . 7 |- (x = <.y, z>. -> (A.y e. A A.z e. B ps -> (x e. (A X. B) -> ph)))
24	18, 23	19.23ai 1064	. . . . . 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 1064	. . . . 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 10	. . . 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 67	. . 3 |- (A.y e. A A.z e. B ps -> (x e. (A X. B) -> ph))
28	27	r19.21aiv 1713	. 2 |- (A.y e. A A.z e. B ps -> A.x e. (A X. B)ph)
29	6, 28	impbi 157	1 |- (A.x e. (A X. B)ph <-> A.y e. A A.z e. B ps)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  A.wral 1645  <.cop 2411   X. cxp 3168

This theorem is referenced by:  rexxp 3219  ralxpf 3220  ffnoprval 4014  eqfnoprval 4016  f1stres 4093  f2ndres 4094  df1st2 4126  df2nd2 4127  rankxplim 4712

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-opab 2667  df-xp 3184

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