Theorem rgen3 1265

Description: Generalization rule for restricted quantification.

Hypothesis

Ref	Expression
rgen3.1	|- ((x e. A /\ y e. A /\ z e. A) -> ph)

Assertion

Ref	Expression
rgen3	|- A.x e. A A.y e. A A.z e. A ph

Distinct variable group(s):   y,z,A   x,z

Proof of Theorem rgen3

Step	Hyp	Ref	Expression
1		rgen3.1	. . . . 5 |- ((x e. A /\ y e. A /\ z e. A) -> ph)
2	1	3exp 611	. . . 4 |- (x e. A -> (y e. A -> (z e. A -> ph)))
3	2	imp 277	. . 3 |- ((x e. A /\ y e. A) -> (z e. A -> ph))
4	3	r19.21aiv 1259	. 2 |- ((x e. A /\ y e. A) -> A.z e. A ph)
5	4	rgen2 1248	1 |- A.x e. A A.y e. A A.z e. A ph

Syntax hints:   -> wi 2   /\ wa 196   /\ w3a 581   e. wcel 1092  A.wral 1201

This theorem is referenced by:  itlso 2151  zorn2 3612

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-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-cleq 1097  df-clel 1099  df-ral 1205

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