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Theorem rexim 2430
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 df-ral 2328 . . . 4  |-  ( A. x  e.  A  ( ph  ->  ps )  <->  A. x
( x  e.  A  ->  ( ph  ->  ps ) ) )
2 simpl 106 . . . . . . 7  |-  ( ( x  e.  A  /\  ph )  ->  x  e.  A )
32a1i 9 . . . . . 6  |-  ( ( x  e.  A  -> 
( ph  ->  ps )
)  ->  ( (
x  e.  A  /\  ph )  ->  x  e.  A ) )
4 pm3.31 253 . . . . . 6  |-  ( ( x  e.  A  -> 
( ph  ->  ps )
)  ->  ( (
x  e.  A  /\  ph )  ->  ps )
)
53, 4jcad 295 . . . . 5  |-  ( ( x  e.  A  -> 
( ph  ->  ps )
)  ->  ( (
x  e.  A  /\  ph )  ->  ( x  e.  A  /\  ps )
) )
65alimi 1360 . . . 4  |-  ( A. x ( x  e.  A  ->  ( ph  ->  ps ) )  ->  A. x ( ( x  e.  A  /\  ph )  ->  ( x  e.  A  /\  ps )
) )
71, 6sylbi 118 . . 3  |-  ( A. x  e.  A  ( ph  ->  ps )  ->  A. x ( ( x  e.  A  /\  ph )  ->  ( x  e.  A  /\  ps )
) )
8 exim 1506 . . 3  |-  ( A. x ( ( x  e.  A  /\  ph )  ->  ( x  e.  A  /\  ps )
)  ->  ( E. x ( x  e.  A  /\  ph )  ->  E. x ( x  e.  A  /\  ps ) ) )
97, 8syl 14 . 2  |-  ( A. x  e.  A  ( ph  ->  ps )  -> 
( E. x ( x  e.  A  /\  ph )  ->  E. x
( x  e.  A  /\  ps ) ) )
10 df-rex 2329 . 2  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
11 df-rex 2329 . 2  |-  ( E. x  e.  A  ps  <->  E. x ( x  e.  A  /\  ps )
)
129, 10, 113imtr4g 198 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:    -> wi 4    /\ wa 101   A.wal 1257   E.wex 1397    e. wcel 1409   A.wral 2323   E.wrex 2324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-ral 2328  df-rex 2329
This theorem is referenced by:  reximia  2431  reximdai  2434  r19.29  2467  reupick2  3251  ss2iun  3700  chfnrn  5306
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