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Theorem r19.37 2925
Description: Restricted quantifier version of one direction of 19.37 2065. (The other direction does not hold when 
A is empty.) (Contributed by FL, 13-May-2012.) (Revised by Mario Carneiro, 11-Dec-2016.)
Hypothesis
Ref Expression
r19.37.1  |-  F/ x ph
Assertion
Ref Expression
r19.37  |-  ( E. x  e.  A  (
ph  ->  ps )  -> 
( ph  ->  E. x  e.  A  ps )
)

Proof of Theorem r19.37
StepHypRef Expression
1 r19.35 2923 . 2  |-  ( E. x  e.  A  (
ph  ->  ps )  <->  ( A. x  e.  A  ph  ->  E. x  e.  A  ps ) )
2 r19.37.1 . . . 4  |-  F/ x ph
3 ax-1 6 . . . 4  |-  ( ph  ->  ( x  e.  A  ->  ph ) )
42, 3ralrimi 2800 . . 3  |-  ( ph  ->  A. x  e.  A  ph )
54imim1i 59 . 2  |-  ( ( A. x  e.  A  ph 
->  E. x  e.  A  ps )  ->  ( ph  ->  E. x  e.  A  ps ) )
61, 5sylbi 200 1  |-  ( E. x  e.  A  (
ph  ->  ps )  -> 
( ph  ->  E. x  e.  A  ps )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   F/wnf 1675    e. wcel 1904   A.wral 2756   E.wrex 2757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-12 1950
This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-nf 1676  df-ral 2761  df-rex 2762
This theorem is referenced by:  r19.37v  2926  ss2iundf  36322
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