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

Theorem rexim 2815
Description: Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Nov-1994.) (Proof shortened by Andrew Salmon, 30-May-2011.)
Assertion
Ref Expression
rexim  |-  ( A. x  e.  A  ( ph  ->  ps )  -> 
( E. x  e.  A  ph  ->  E. x  e.  A  ps )
)

Proof of Theorem rexim
StepHypRef Expression
1 con3 134 . . . 4  |-  ( (
ph  ->  ps )  -> 
( -.  ps  ->  -. 
ph ) )
21ral2imi 2787 . . 3  |-  ( A. x  e.  A  ( ph  ->  ps )  -> 
( A. x  e.  A  -.  ps  ->  A. x  e.  A  -.  ph ) )
32con3d 133 . 2  |-  ( A. x  e.  A  ( ph  ->  ps )  -> 
( -.  A. x  e.  A  -.  ph  ->  -. 
A. x  e.  A  -.  ps ) )
4 dfrex2 2723 . 2  |-  ( E. x  e.  A  ph  <->  -. 
A. x  e.  A  -.  ph )
5 dfrex2 2723 . 2  |-  ( E. x  e.  A  ps  <->  -. 
A. x  e.  A  -.  ps )
63, 4, 53imtr4g 270 1  |-  ( A. x  e.  A  ( ph  ->  ps )  -> 
( E. x  e.  A  ph  ->  E. x  e.  A  ps )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wral 2710   E.wrex 2711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1587  df-ral 2715  df-rex 2716
This theorem is referenced by:  reximia  2816  reximdai  2819  r19.29  2852  reupick2  3631  ss2iun  4181  chfnrn  5809  isf32lem2  8515  ptcmplem4  19602  bnj110  31738
  Copyright terms: Public domain W3C validator