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Theorem rgen3 2640
Description: Generalization rule for restricted quantification. (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 1151 . . 3  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  z  e.  C )  ->  ph )
32ralrimiva 2626 . 2  |-  ( ( x  e.  A  /\  y  e.  B )  ->  A. z  e.  C  ph )
43rgen2 2639 1  |-  A. x  e.  A  A. y  e.  B  A. z  e.  C  ph
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    e. wcel 1684   A.wral 2543
This theorem is referenced by:  isposi  14090  addcnlem  18368  isgrpoi  20865  cnrngo  21070  lnocoi  21335  0lnfn  22565  lnopcoi  22583  poseq  24253  1cat  25759
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-nf 1532  df-ral 2548
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