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Theorem rgen3 2808
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
StepHypRef Expression
1 rgen3.1 . . . 4  |-  ( ( x  e.  A  /\  y  e.  B  /\  z  e.  C )  ->  ph )
213expa 1194 . . 3  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  z  e.  C )  ->  ph )
32ralrimiva 2796 . 2  |-  ( ( x  e.  A  /\  y  e.  B )  ->  A. z  e.  C  ph )
43rgen2 2807 1  |-  A. x  e.  A  A. y  e.  B  A. z  e.  C  ph
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    e. wcel 1826   A.wral 2732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712
This theorem depends on definitions:  df-bi 185  df-an 369  df-3an 973  df-ral 2737
This theorem is referenced by:  isposi  15703  addcnlem  21453  isgrpoi  25317  cnrngo  25522  lnocoi  25789  0lnfn  27020  lnopcoi  27038  xrge0omnd  27854  reofld  27984  poseq  29498  2zrngasgrp  32946  2zrngmsgrp  32953  2zrngALT  32954
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