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Theorem ralm 3353
Description: Inhabited classes and restricted quantification. (Contributed by Jim Kingdon, 6-Aug-2018.)
Assertion
Ref Expression
ralm  |-  ( ( E. x  x  e.  A  ->  A. x  e.  A  ph )  <->  A. x  e.  A  ph )

Proof of Theorem ralm
StepHypRef Expression
1 df-ral 2328 . . . . . 6  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
21imbi2i 219 . . . . 5  |-  ( ( E. x  x  e.  A  ->  A. x  e.  A  ph )  <->  ( E. x  x  e.  A  ->  A. x ( x  e.  A  ->  ph )
) )
3 19.38 1582 . . . . 5  |-  ( ( E. x  x  e.  A  ->  A. x
( x  e.  A  ->  ph ) )  ->  A. x ( x  e.  A  ->  ( x  e.  A  ->  ph )
) )
42, 3sylbi 118 . . . 4  |-  ( ( E. x  x  e.  A  ->  A. x  e.  A  ph )  ->  A. x ( x  e.  A  ->  ( x  e.  A  ->  ph )
) )
5 pm2.43 51 . . . . 5  |-  ( ( x  e.  A  -> 
( x  e.  A  ->  ph ) )  -> 
( x  e.  A  ->  ph ) )
65alimi 1360 . . . 4  |-  ( A. x ( x  e.  A  ->  ( x  e.  A  ->  ph )
)  ->  A. x
( x  e.  A  ->  ph ) )
74, 6syl 14 . . 3  |-  ( ( E. x  x  e.  A  ->  A. x  e.  A  ph )  ->  A. x ( x  e.  A  ->  ph ) )
87, 1sylibr 141 . 2  |-  ( ( E. x  x  e.  A  ->  A. x  e.  A  ph )  ->  A. x  e.  A  ph )
9 ax-1 5 . 2  |-  ( A. x  e.  A  ph  ->  ( E. x  x  e.  A  ->  A. x  e.  A  ph ) )
108, 9impbii 121 1  |-  ( ( E. x  x  e.  A  ->  A. x  e.  A  ph )  <->  A. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 102   A.wal 1257   E.wex 1397    e. wcel 1409   A.wral 2323
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  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-ral 2328
This theorem is referenced by:  raaan  3355
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