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Theorem r19.37 2651
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 2649 . 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 7 . . . 4  |-  ( ph  ->  ( x  e.  A  ->  ph ) )
42, 3ralrimi 2586 . . 3  |-  ( ph  ->  A. x  e.  A  ph )
54imim1i 56 . 2  |-  ( ( A. x  e.  A  ph 
->  E. x  e.  A  ps )  ->  ( ph  ->  E. x  e.  A  ps ) )
61, 5sylbi 189 1  |-  ( E. x  e.  A  (
ph  ->  ps )  -> 
( ph  ->  E. x  e.  A  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 6   F/wnf 1539    e. wcel 1621   A.wral 2509   E.wrex 2510
This theorem is referenced by:  r19.37av  2652
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-gen 1536  ax-17 1628  ax-4 1692
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-ral 2513  df-rex 2514
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