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Theorem rexm 3348
Description: Restricted existential quantification implies its restriction is inhabited. (Contributed by Jim Kingdon, 16-Oct-2018.)
Assertion
Ref Expression
rexm  |-  ( E. x  e.  A  ph  ->  E. x  x  e.  A )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem rexm
StepHypRef Expression
1 df-rex 2329 . 2  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
2 simpl 106 . . 3  |-  ( ( x  e.  A  /\  ph )  ->  x  e.  A )
32eximi 1507 . 2  |-  ( E. x ( x  e.  A  /\  ph )  ->  E. x  x  e.  A )
41, 3sylbi 118 1  |-  ( E. x  e.  A  ph  ->  E. x  x  e.  A )
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-ial 1443
This theorem depends on definitions:  df-bi 114  df-rex 2329
This theorem is referenced by:  eusvobj2  5526
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