MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  19.9t Structured version   Unicode version

Theorem 19.9t 1838
Description: A closed version of 19.9 1841. (Contributed by NM, 13-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)
Assertion
Ref Expression
19.9t  |-  ( F/ x ph  ->  ( E. x ph  <->  ph ) )

Proof of Theorem 19.9t
StepHypRef Expression
1 df-nf 1600 . . 3  |-  ( F/ x ph  <->  A. x
( ph  ->  A. x ph ) )
2 19.9ht 1837 . . 3  |-  ( A. x ( ph  ->  A. x ph )  -> 
( E. x ph  ->  ph ) )
31, 2sylbi 195 . 2  |-  ( F/ x ph  ->  ( E. x ph  ->  ph )
)
4 19.8a 1806 . 2  |-  ( ph  ->  E. x ph )
53, 4impbid1 203 1  |-  ( F/ x ph  ->  ( E. x ph  <->  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1377   E.wex 1596   F/wnf 1599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-12 1803
This theorem depends on definitions:  df-bi 185  df-ex 1597  df-nf 1600
This theorem is referenced by:  19.9h  1839  19.9d  1840  19.21t  1852  19.23t  1856  spimt  1974  sbft  2093  vtoclegft  3190  exlimddvf  30453  bj-cbv3tb  33754  bj-spimtv  33761  bj-sbftv  33827  bj-equsal1t  33877  bj-19.21t  33885
  Copyright terms: Public domain W3C validator