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Theorem rexim 2778
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 2750 . . 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 2687 . 2  |-  ( E. x  e.  A  ph  <->  -. 
A. x  e.  A  -.  ph )
5 dfrex2 2687 . 2  |-  ( E. x  e.  A  ps  <->  -. 
A. x  e.  A  -.  ps )
63, 4, 53imtr4g 262 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 2674   E.wrex 2675
This theorem is referenced by:  reximia  2779  reximdai  2782  r19.29  2814  reupick2  3595  ss2iun  4076  chfnrn  5808  isf32lem2  8198  ptcmplem4  18047  bnj110  28947
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-ral 2679  df-rex 2680
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