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Theorem reximdva0m 3409
Description: Restricted existence deduced from inhabited class. (Contributed by Jim Kingdon, 31-Jul-2018.)
Hypothesis
Ref Expression
reximdva0m.1  |-  ( (
ph  /\  x  e.  A )  ->  ps )
Assertion
Ref Expression
reximdva0m  |-  ( (
ph  /\  E. x  x  e.  A )  ->  E. x  e.  A  ps )
Distinct variable groups:    x, A    ph, x
Allowed substitution hint:    ps( x)

Proof of Theorem reximdva0m
StepHypRef Expression
1 reximdva0m.1 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  ps )
21ex 114 . . . . 5  |-  ( ph  ->  ( x  e.  A  ->  ps ) )
32ancld 323 . . . 4  |-  ( ph  ->  ( x  e.  A  ->  ( x  e.  A  /\  ps ) ) )
43eximdv 1860 . . 3  |-  ( ph  ->  ( E. x  x  e.  A  ->  E. x
( x  e.  A  /\  ps ) ) )
54imp 123 . 2  |-  ( (
ph  /\  E. x  x  e.  A )  ->  E. x ( x  e.  A  /\  ps ) )
6 df-rex 2441 . 2  |-  ( E. x  e.  A  ps  <->  E. x ( x  e.  A  /\  ps )
)
75, 6sylibr 133 1  |-  ( (
ph  /\  E. x  x  e.  A )  ->  E. x  e.  A  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   E.wex 1472    e. wcel 2128   E.wrex 2436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-17 1506  ax-ial 1514
This theorem depends on definitions:  df-bi 116  df-rex 2441
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator