Theorem rgen3 2976

Description: Generalization rule for restricted quantification, with three quantifiers. (Contributed by NM, 12-Jan-2008.)

Hypothesis

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

Assertion

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

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

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

Proof of Theorem rgen3

Step	Hyp	Ref	Expression
1		rgen3.1	. . . 4 |- ( ( x e. A /\ y e. B /\ z e. C ) -> ph )
2	1	3expa 1265	. . 3 |- ( ( ( x e. A /\ y e. B ) /\ z e. C ) -> ph )
3	2	ralrimiva 2966	. 2 |- ( ( x e. A /\ y e. B ) -> A. z e. C ph )
4	3	rgen2 2975	1 |- A. x e. A A. y e. B A. z e. C ph

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   e. wcel 1990   A.wral 2912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839

This theorem depends on definitions:  df-bi 197  df-an 386  df-3an 1039  df-ral 2917

This theorem is referenced by:  isposi  16956  addcnlem  22667  isgrpoi  27352  lnocoi  27612  0lnfn  28844  lnopcoi  28862  xrge0omnd  29711  reofld  29840  poseq  31750  2zrngasgrp  41940  2zrngmsgrp  41947  2zrngALT  41948