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Theorem rexim 2829
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 139 . . . 4  |-  ( (
ph  ->  ps )  -> 
( -.  ps  ->  -. 
ph ) )
21ral2imi 2753 . . 3  |-  ( A. x  e.  A  ( ph  ->  ps )  -> 
( A. x  e.  A  -.  ps  ->  A. x  e.  A  -.  ph ) )
32con3d 138 . 2  |-  ( A. x  e.  A  ( ph  ->  ps )  -> 
( -.  A. x  e.  A  -.  ph  ->  -. 
A. x  e.  A  -.  ps ) )
4 dfrex2 2815 . 2  |-  ( E. x  e.  A  ph  <->  -. 
A. x  e.  A  -.  ph )
5 dfrex2 2815 . 2  |-  ( E. x  e.  A  ps  <->  -. 
A. x  e.  A  -.  ps )
63, 4, 53imtr4g 273 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 2714   E.wrex 2715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1658  df-ral 2719  df-rex 2720
This theorem is referenced by:  reximia  2830  reximdai  2833  reximdvai  2836  r19.29  2902  reupick2  3702  ss2iun  4258  chfnrn  5952  isf32lem2  8735  ptcmplem4  21012  bnj110  29621  poimirlem25  31872
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