Theorem ralxp 2456

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 group(s):   x,y,z,A   x,B,y,z   ph,y,z   ps,x

Proof of Theorem ralxp

Step	Hyp	Ref	Expression
1		ralxp.1	. . . . . . 7 |- (x = <.y, z>. -> (ph <-> ps))
2	1	rcla4v 1402	. . . . . 6 |- (A.x e. (A X. B)ph -> (<.y, z>. e. (A X. B) -> ps))
3		visset 1350	. . . . . . 7 |- z e. V
4	3	opelxp 2452	. . . . . 6 |- (<.y, z>. e. (A X. B) <-> (y e. A /\ z e. B))
5	2, 4	syl5ibr 182	. . . . 5 |- (A.x e. (A X. B)ph -> ((y e. A /\ z e. B) -> ps))
6	5	exp3a 292	. . . 4 |- (A.x e. (A X. B)ph -> (y e. A -> (z e. B -> ps)))
7	6	r19.21adv 1262	. . 3 |- (A.x e. (A X. B)ph -> (y e. A -> A.z e. B ps))
8	7	r19.21aiv 1259	. 2 |- (A.x e. (A X. B)ph -> A.y e. A A.z e. B ps)
9		elxp 2442	. . . . . 6 |- (x e. (A X. B) <-> E.yE.z(x = <.y, z>. /\ (y e. A /\ z e. B)))
10		pm3.26 256	. . . . . . . 8 |- ((x = <.y, z>. /\ (y e. A /\ z e. B)) -> x = <.y, z>.)
11	10	19.22i 723	. . . . . . 7 |- (E.z(x = <.y, z>. /\ (y e. A /\ z e. B)) -> E.z x = <.y, z>.)
12	11	19.22i 723	. . . . . 6 |- (E.yE.z(x = <.y, z>. /\ (y e. A /\ z e. B)) -> E.yE.z x = <.y, z>.)
13	9, 12	sylbi 174	. . . . 5 |- (x e. (A X. B) -> E.yE.z x = <.y, z>.)
14		hbra1 1237	. . . . . . 7 |- (A.y e. A A.z e. B ps -> A.yA.y e. A A.z e. B ps)
15		ax-17 925	. . . . . . 7 |- ((x e. (A X. B) -> ph) -> A.y(x e. (A X. B) -> ph))
16	14, 15	hbim 702	. . . . . 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)))
17		ax-17 925	. . . . . . . . 9 |- (y e. A -> A.z y e. A)
18		hbra1 1237	. . . . . . . . 9 |- (A.z e. B ps -> A.zA.z e. B ps)
19	17, 18	hbral 1236	. . . . . . . 8 |- (A.y e. A A.z e. B ps -> A.zA.y e. A A.z e. B ps)
20		ax-17 925	. . . . . . . 8 |- ((x e. (A X. B) -> ph) -> A.z(x e. (A X. B) -> ph))
21	19, 20	hbim 702	. . . . . . 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)))
22		eleq1 1149	. . . . . . . . . 10 |- (x = <.y, z>. -> (x e. (A X. B) <-> <.y, z>. e. (A X. B)))
23	22, 4	syl6bb 414	. . . . . . . . 9 |- (x = <.y, z>. -> (x e. (A X. B) <-> (y e. A /\ z e. B)))
24	23, 1	imbi12d 474	. . . . . . . 8 |- (x = <.y, z>. -> ((x e. (A X. B) -> ph) <-> ((y e. A /\ z e. B) -> ps)))
25		ra42 1245	. . . . . . . 8 |- (A.y e. A A.z e. B ps -> ((y e. A /\ z e. B) -> ps))
26	24, 25	syl5bir 184	. . . . . . 7 |- (x = <.y, z>. -> (A.y e. A A.z e. B ps -> (x e. (A X. B) -> ph)))
27	21, 26	19.23ai 746	. . . . . 6 |- (E.z x = <.y, z>. -> (A.y e. A A.z e. B ps -> (x e. (A X. B) -> ph)))
28	16, 27	19.23ai 746	. . . . 5 |- (E.yE.z x = <.y, z>. -> (A.y e. A A.z e. B ps -> (x e. (A X. B) -> ph)))
29	13, 28	syl 12	. . . 4 |- (x e. (A X. B) -> (A.y e. A A.z e. B ps -> (x e. (A X. B) -> ph)))
30	29	pm2.43b 61	. . 3 |- (A.y e. A A.z e. B ps -> (x e. (A X. B) -> ph))
31	30	r19.21aiv 1259	. 2 |- (A.y e. A A.z e. B ps -> A.x e. (A X. B)ph)
32	8, 31	impbi 139	1 |- (A.x e. (A X. B)ph <-> A.y e. A A.z e. B ps)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  E.wex 678   = wceq 1091   e. wcel 1092  A.wral 1201  <.cop 1810   X. cxp 2408

This theorem is referenced by:  ffnoprval 3041  f1stres 3096  df1st2 3098

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-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098  df-xp 2424

metamath.org