Theorem ralxp 3213

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 1870	. . . 4 |- (A.x e. (A X. B)ph -> (<.y, z>. e. (A X. B) -> ps))
3		visset 1809	. . . . 5 |- z e. V
4	3	opelxp 3209	. . . 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 1717	. 2 |- (A.x e. (A X. B)ph -> A.y e. A A.z e. B ps)
7		elxp 3197	. . . . . 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 1039	. . . . . 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 1684	. . . . . . 7 |- (A.y e. A A.z e. B ps -> A.yA.y e. A A.z e. B ps)
12		ax-17 969	. . . . . . 7 |- ((x e. (A X. B) -> ph) -> A.y(x e. (A X. B) -> ph))
13	11, 12	hbim 1005	. . . . . 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-17 969	. . . . . . . . 9 |- (y e. A -> A.z y e. A)
15		hbra1 1684	. . . . . . . . 9 |- (A.z e. B ps -> A.zA.z e. B ps)
16	14, 15	hbral 1683	. . . . . . . 8 |- (A.y e. A A.z e. B ps -> A.zA.y e. A A.z e. B ps)
17		ax-17 969	. . . . . . . 8 |- ((x e. (A X. B) -> ph) -> A.z(x e. (A X. B) -> ph))
18	16, 17	hbim 1005	. . . . . . 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 1531	. . . . . . . . . 10 |- (x = <.y, z>. -> (x e. (A X. B) <-> <.y, z>. e. (A X. B)))
20	19, 4	syl6bb 535	. . . . . . . . 9 |- (x = <.y, z>. -> (x e. (A X. B) <-> (y e. A /\ z e. B)))
21	20, 1	imbi12d 625	. . . . . . . 8 |- (x = <.y, z>. -> ((x e. (A X. B) -> ph) <-> ((y e. A /\ z e. B) -> ps)))
22		ra42 1693	. . . . . . . 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 1062	. . . . . 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 1062	. . . . 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 1710	. 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 954   e. wcel 956  E.wex 978  A.wral 1642  <.cop 2407   X. cxp 3163

This theorem is referenced by:  rexxp 3214  ralxpf 3215  ffnoprval 4005  eqfnoprval 4007  f1stres 4083  f2ndres 4084  df1st2 4116  df2nd2 4117  rankxplim 4692

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-opab 2662  df-xp 3179

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