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Theorem rexim 2648
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 128 . . . 4  |-  ( (
ph  ->  ps )  -> 
( -.  ps  ->  -. 
ph ) )
21ral2imi 2620 . . 3  |-  ( A. x  e.  A  ( ph  ->  ps )  -> 
( A. x  e.  A  -.  ps  ->  A. x  e.  A  -.  ph ) )
32con3d 127 . 2  |-  ( A. x  e.  A  ( ph  ->  ps )  -> 
( -.  A. x  e.  A  -.  ph  ->  -. 
A. x  e.  A  -.  ps ) )
4 dfrex2 2557 . 2  |-  ( E. x  e.  A  ph  <->  -. 
A. x  e.  A  -.  ph )
5 dfrex2 2557 . 2  |-  ( E. x  e.  A  ps  <->  -. 
A. x  e.  A  -.  ps )
63, 4, 53imtr4g 263 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 5    -> wi 6   A.wral 2544   E.wrex 2545
This theorem is referenced by:  reximia  2649  reximdai  2652  r19.29  2684  reupick2  3455  ss2iun  3921  chfnrn  5597  isf32lem2  7975  ptcmplem4  17743  dstr  24570  bnj110  28157
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1534  df-ral 2549  df-rex 2550
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