MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  alral Structured version   Unicode version

Theorem alral 2756
Description: Universal quantification implies restricted quantification. (Contributed by NM, 20-Oct-2006.)
Assertion
Ref Expression
alral  |-  ( A. x ph  ->  A. x  e.  A  ph )

Proof of Theorem alral
StepHypRef Expression
1 ax-1 5 . . 3  |-  ( ph  ->  ( x  e.  A  ->  ph ) )
21alimi 1568 . 2  |-  ( A. x ph  ->  A. x
( x  e.  A  ->  ph ) )
3 df-ral 2702 . 2  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
42, 3sylibr 204 1  |-  ( A. x ph  ->  A. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1549    e. wcel 1725   A.wral 2697
This theorem is referenced by:  find  4862  brdom5  8399  brdom4  8400  sumeq2w  12478  rpnnen2  12817  prodeq2w  25230  elpotr  25400  rexrsb  27904  ordelordALTVD  28906
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566
This theorem depends on definitions:  df-bi 178  df-ral 2702
  Copyright terms: Public domain W3C validator