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Theorem rexim 2647
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 126 . . . 4  |-  ( (
ph  ->  ps )  -> 
( -.  ps  ->  -. 
ph ) )
21ral2imi 2619 . . 3  |-  ( A. x  e.  A  ( ph  ->  ps )  -> 
( A. x  e.  A  -.  ps  ->  A. x  e.  A  -.  ph ) )
32con3d 125 . 2  |-  ( A. x  e.  A  ( ph  ->  ps )  -> 
( -.  A. x  e.  A  -.  ph  ->  -. 
A. x  e.  A  -.  ps ) )
4 dfrex2 2556 . 2  |-  ( E. x  e.  A  ph  <->  -. 
A. x  e.  A  -.  ph )
5 dfrex2 2556 . 2  |-  ( E. x  e.  A  ps  <->  -. 
A. x  e.  A  -.  ps )
63, 4, 53imtr4g 261 1  |-  ( A. x  e.  A  ( ph  ->  ps )  -> 
( E. x  e.  A  ph  ->  E. x  e.  A  ps )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wral 2543   E.wrex 2544
This theorem is referenced by:  reximia  2648  reximdai  2651  r19.29  2683  reupick2  3454  ss2iun  3920  chfnrn  5636  isf32lem2  7980  ptcmplem4  17749  dstr  25171  bnj110  28890
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-ral 2548  df-rex 2549
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