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Theorem r19.37 2867
Description: Restricted version of one direction of Theorem 19.37 of [Margaris] p. 90. (The other direction doesn't 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 2865 . 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 2795 . . 3  |-  ( ph  ->  A. x  e.  A  ph )
54imim1i 58 . 2  |-  ( ( A. x  e.  A  ph 
->  E. x  e.  A  ps )  ->  ( ph  ->  E. x  e.  A  ps ) )
61, 5sylbi 195 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 1594    e. wcel 1761   A.wral 2713   E.wrex 2714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-12 1797
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1367  df-ex 1592  df-nf 1595  df-ral 2718  df-rex 2719
This theorem is referenced by:  r19.37av  2868
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