Theorem bnd2 3917

Description: A variant of the Boundedness Axiom bnd 3916 that picks a subset z out of a possibly proper class B in which a property is true. (Contributed by NM, 4-Feb-2004.)

Hypothesis

Ref	Expression
bnd2.1	|- A e. _V

Assertion

Ref	Expression
bnd2	|- (A.x e. A E.y e. B ph -> E.z(z C_ B /\ A.x e. A E.y e. z ph))

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

Allowed substitution hints:   ph(x,y)   A(y)

Proof of Theorem bnd2

Dummy variables w v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rex 2306	. . . 4 |- (E.y e. B ph <-> E.y(y e. B /\ ph))
2	1	ralbii 2324	. . 3 |- (A.x e. A E.y e. B ph <-> A.x e. A E.y(y e. B /\ ph))
3		bnd2.1	. . . 4 |- A e. _V
4		raleq 2499	. . . . 5 |- (v = A -> (A.x e. v E.y(y e. B /\ ph) <-> A.x e. A E.y(y e. B /\ ph)))
5		raleq 2499	. . . . . 6 |- (v = A -> (A.x e. v E.y e. w (y e. B /\ ph) <-> A.x e. A E.y e. w (y e. B /\ ph)))
6	5	exbidv 1703	. . . . 5 |- (v = A -> (E.wA.x e. v E.y e. w (y e. B /\ ph) <-> E.wA.x e. A E.y e. w (y e. B /\ ph)))
7	4, 6	imbi12d 223	. . . 4 |- (v = A -> ((A.x e. v E.y(y e. B /\ ph) -> E.wA.x e. v E.y e. w (y e. B /\ ph)) <-> (A.x e. A E.y(y e. B /\ ph) -> E.wA.x e. A E.y e. w (y e. B /\ ph))))
8		bnd 3916	. . . 4 |- (A.x e. v E.y(y e. B /\ ph) -> E.wA.x e. v E.y e. w (y e. B /\ ph))
9	3, 7, 8	vtocl 2602	. . 3 |- (A.x e. A E.y(y e. B /\ ph) -> E.wA.x e. A E.y e. w (y e. B /\ ph))
10	2, 9	sylbi 114	. 2 |- (A.x e. A E.y e. B ph -> E.wA.x e. A E.y e. w (y e. B /\ ph))
11		vex 2554	. . . . 5 |- w e. _V
12	11	inex1 3882	. . . 4 |- (w i^i B) e. _V
13		inss2 3152	. . . . . . 7 |- (w i^i B) C_ B
14		sseq1 2960	. . . . . . 7 |- (z = (w i^i B) -> (z C_ B <-> (w i^i B) C_ B))
15	13, 14	mpbiri 157	. . . . . 6 |- (z = (w i^i B) -> z C_ B)
16	15	biantrurd 289	. . . . 5 |- (z = (w i^i B) -> (A.x e. A E.y e. z ph <-> (z C_ B /\ A.x e. A E.y e. z ph)))
17		rexeq 2500	. . . . . . 7 |- (z = (w i^i B) -> (E.y e. z ph <-> E.y e. (w i^i B)ph))
18		elin 3120	. . . . . . . . . 10 |- (y e. (w i^i B) <-> (y e. w /\ y e. B))
19	18	anbi1i 431	. . . . . . . . 9 |- ((y e. (w i^i B) /\ ph) <-> ((y e. w /\ y e. B) /\ ph))
20		anass 381	. . . . . . . . 9 |- (((y e. w /\ y e. B) /\ ph) <-> (y e. w /\ (y e. B /\ ph)))
21	19, 20	bitri 173	. . . . . . . 8 |- ((y e. (w i^i B) /\ ph) <-> (y e. w /\ (y e. B /\ ph)))
22	21	rexbii2 2329	. . . . . . 7 |- (E.y e. (w i^i B)ph <-> E.y e. w (y e. B /\ ph))
23	17, 22	syl6bb 185	. . . . . 6 |- (z = (w i^i B) -> (E.y e. z ph <-> E.y e. w (y e. B /\ ph)))
24	23	ralbidv 2320	. . . . 5 |- (z = (w i^i B) -> (A.x e. A E.y e. z ph <-> A.x e. A E.y e. w (y e. B /\ ph)))
25	16, 24	bitr3d 179	. . . 4 |- (z = (w i^i B) -> ((z C_ B /\ A.x e. A E.y e. z ph) <-> A.x e. A E.y e. w (y e. B /\ ph)))
26	12, 25	spcev 2641	. . 3 |- (A.x e. A E.y e. w (y e. B /\ ph) -> E.z(z C_ B /\ A.x e. A E.y e. z ph))
27	26	exlimiv 1486	. 2 |- (E.wA.x e. A E.y e. w (y e. B /\ ph) -> E.z(z C_ B /\ A.x e. A E.y e. z ph))
28	10, 27	syl 14	1 |- (A.x e. A E.y e. B ph -> E.z(z C_ B /\ A.x e. A E.y e. z ph))

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1242  E.wex 1378   e. wcel 1390  A.wral 2300  E.wrex 2301   _Vcvv 2551   i^i cin 2910   C_ wss 2911

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925

This theorem is referenced by: (None)