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Theorem r19.36v 2949
Description: Restricted quantifier version of one direction of 19.36 2054. (The other direction holds iff  A is nonempty, see r19.36zv 3881.) (Contributed by NM, 22-Oct-2003.)
Assertion
Ref Expression
r19.36v  |-  ( E. x  e.  A  (
ph  ->  ps )  -> 
( A. x  e.  A  ph  ->  ps ) )
Distinct variable group:    ps, x
Allowed substitution hints:    ph( x)    A( x)

Proof of Theorem r19.36v
StepHypRef Expression
1 r19.35 2948 . 2  |-  ( E. x  e.  A  (
ph  ->  ps )  <->  ( A. x  e.  A  ph  ->  E. x  e.  A  ps ) )
2 idd 25 . . . 4  |-  ( x  e.  A  ->  ( ps  ->  ps ) )
32rexlimiv 2884 . . 3  |-  ( E. x  e.  A  ps  ->  ps )
43imim2i 16 . 2  |-  ( ( A. x  e.  A  ph 
->  E. x  e.  A  ps )  ->  ( A. x  e.  A  ph  ->  ps ) )
51, 4sylbi 200 1  |-  ( E. x  e.  A  (
ph  ->  ps )  -> 
( A. x  e.  A  ph  ->  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1897   A.wral 2748   E.wrex 2749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1674  df-ral 2753  df-rex 2754
This theorem is referenced by:  iinss  4342  uniimadom  8994  hashgt12el  12627
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