Theorem r3al 1690

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 groups:   x,y,z   y,A,z   z,B

Proof of Theorem r3al

Step	Hyp	Ref	Expression
1		df-ral 1649	. 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 1676	. . 3 |- (A.y e. B A.z e. C ph <-> A.yA.z((y e. B /\ z e. C) -> ph))
3	2	ralbii 1667	. 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 779	. . . . . . . . 9 |- ((x e. A /\ y e. B /\ z e. C) <-> (x e. A /\ (y e. B /\ z e. C)))
5	4	imbi1i 186	. . . . . . . 8 |- (((x e. A /\ y e. B /\ z e. C) -> ph) <-> ((x e. A /\ (y e. B /\ z e. C)) -> ph))
6		impexp 347	. . . . . . . 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 173	. . . . . . 7 |- (((x e. A /\ y e. B /\ z e. C) -> ph) <-> (x e. A -> ((y e. B /\ z e. C) -> ph)))
8	7	albii 999	. . . . . 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 1285	. . . . . 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 173	. . . . 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 999	. . . 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 1285	. . . 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 173	. . 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 999	. 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 183	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 3   <-> wb 146   /\ wa 223   /\ w3a 775  A.wal 954   e. wcel 958  A.wral 1645

This theorem is referenced by:  poss 2841  pocl 2844  dfwe2 2935  cmpmon 10743

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 777  df-ral 1649

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