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Theorem reximdva0m 3264
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 112 . . . . 5  |-  ( ph  ->  ( x  e.  A  ->  ps ) )
32ancld 312 . . . 4  |-  ( ph  ->  ( x  e.  A  ->  ( x  e.  A  /\  ps ) ) )
43eximdv 1776 . . 3  |-  ( ph  ->  ( E. x  x  e.  A  ->  E. x
( x  e.  A  /\  ps ) ) )
54imp 119 . 2  |-  ( (
ph  /\  E. x  x  e.  A )  ->  E. x ( x  e.  A  /\  ps ) )
6 df-rex 2329 . 2  |-  ( E. x  e.  A  ps  <->  E. x ( x  e.  A  /\  ps )
)
75, 6sylibr 141 1  |-  ( (
ph  /\  E. x  x  e.  A )  ->  E. x  e.  A  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101   E.wex 1397    e. wcel 1409   E.wrex 2324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-rex 2329
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator