Theorem r3al 1240

Description: Triple restricted universal quantification.

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 group(s):   x,y,z   y,A,z   z,B

Proof of Theorem r3al

Step	Hyp	Ref	Expression
1		df-ral 1205	. 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 1231	. . 3 |- (A.y e. B A.z e. C ph <-> A.yA.z((y e. B /\ z e. C) -> ph))
3	2	biral 1223	. 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 585	. . . . . . . . 9 |- ((x e. A /\ y e. B /\ z e. C) <-> (x e. A /\ (y e. B /\ z e. C)))
5	4	imbi1i 161	. . . . . . . 8 |- (((x e. A /\ y e. B /\ z e. C) -> ph) <-> ((x e. A /\ (y e. B /\ z e. C)) -> ph))
6		impexp 276	. . . . . . . 8 |- (((x e. A /\ (y e. B /\ z e. C)) -> ph) <-> (x e. A -> ((y e. B /\ z e. C) -> ph)))
7	5, 6	bitr 151	. . . . . . 7 |- (((x e. A /\ y e. B /\ z e. C) -> ph) <-> (x e. A -> ((y e. B /\ z e. C) -> ph)))
8	7	bial 695	. . . . . 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 942	. . . . . 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	bitr 151	. . . . 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	bial 695	. . . 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 942	. . . 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	bitr 151	. . 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	bial 695	. 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	3bitr4 158	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 2   <-> wb 127   /\ wa 196   /\ w3a 581  A.wal 672   e. wcel 1092  A.wral 1201

This theorem is referenced by:  poss 2129  pocl 2132  dfwe2 2187

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-gen 677  ax-17 925

This theorem depends on definitions:  df-bi 128  df-an 198  df-3an 583  df-ral 1205

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