Theorem r3al 2672

Description: Triple restricted universal quantification. (Contributed by NM, 19-Nov-1995.)

Assertion

Ref	Expression
r3al	|- (A.x e. A A.y e. B A.z e. C ph <-> A.xA.yA.z((x e. A /\ y e. B /\ z e. C) -> ph))

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

Allowed substitution hints:   ph(x,y,z)   A(x)   B(x,y)   C(x,y,z)

Proof of Theorem r3al

Step	Hyp	Ref	Expression
1		df-ral 2620	. 2 |- (A.x e. A A.yA.z((y e. B /\ z e. C) -> ph) <-> A.x(x e. A -> A.yA.z((y e. B /\ z e. C) -> ph)))
2		r2al 2652	. . 3 |- (A.y e. B A.z e. C ph <-> A.yA.z((y e. B /\ z e. C) -> ph))
3	2	ralbii 2639	. 2 |- (A.x e. A A.y e. B A.z e. C ph <-> A.x e. A A.yA.z((y e. B /\ z e. C) -> ph))
4		3anass 938	. . . . . . . . 9 |- ((x e. A /\ y e. B /\ z e. C) <-> (x e. A /\ (y e. B /\ z e. C)))
5	4	imbi1i 315	. . . . . . . 8 |- (((x e. A /\ y e. B /\ z e. C) -> ph) <-> ((x e. A /\ (y e. B /\ z e. C)) -> ph))
6		impexp 433	. . . . . . . 8 |- (((x e. A /\ (y e. B /\ z e. C)) -> ph) <-> (x e. A -> ((y e. B /\ z e. C) -> ph)))
7	5, 6	bitri 240	. . . . . . 7 |- (((x e. A /\ y e. B /\ z e. C) -> ph) <-> (x e. A -> ((y e. B /\ z e. C) -> ph)))
8	7	albii 1566	. . . . . 6 |- (A.z((x e. A /\ y e. B /\ z e. C) -> ph) <-> A.z(x e. A -> ((y e. B /\ z e. C) -> ph)))
9		19.21v 1890	. . . . . 6 |- (A.z(x e. A -> ((y e. B /\ z e. C) -> ph)) <-> (x e. A -> A.z((y e. B /\ z e. C) -> ph)))
10	8, 9	bitri 240	. . . . 5 |- (A.z((x e. A /\ y e. B /\ z e. C) -> ph) <-> (x e. A -> A.z((y e. B /\ z e. C) -> ph)))
11	10	albii 1566	. . . 4 |- (A.yA.z((x e. A /\ y e. B /\ z e. C) -> ph) <-> A.y(x e. A -> A.z((y e. B /\ z e. C) -> ph)))
12		19.21v 1890	. . . 4 |- (A.y(x e. A -> A.z((y e. B /\ z e. C) -> ph)) <-> (x e. A -> A.yA.z((y e. B /\ z e. C) -> ph)))
13	11, 12	bitri 240	. . 3 |- (A.yA.z((x e. A /\ y e. B /\ z e. C) -> ph) <-> (x e. A -> A.yA.z((y e. B /\ z e. C) -> ph)))
14	13	albii 1566	. 2 |- (A.xA.yA.z((x e. A /\ y e. B /\ z e. C) -> ph) <-> A.x(x e. A -> A.yA.z((y e. B /\ z e. C) -> ph)))
15	1, 3, 14	3bitr4i 268	1 |- (A.x e. A A.y e. B A.z e. C ph <-> A.xA.yA.z((x e. A /\ y e. B /\ z e. C) -> ph))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934  A.wal 1540   e. wcel 1710  A.wral 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620

This theorem is referenced by: (None)