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

Theorem 19.9t 1795
Description: A closed version of 19.9 1799. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortended 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 1555 . . 3  |-  ( F/ x ph  <->  A. x
( ph  ->  A. x ph ) )
2 19.9ht 1794 . . 3  |-  ( A. x ( ph  ->  A. x ph )  -> 
( E. x ph  ->  ph ) )
31, 2sylbi 189 . 2  |-  ( F/ x ph  ->  ( E. x ph  ->  ph )
)
4 19.8a 1764 . 2  |-  ( ph  ->  E. x ph )
53, 4impbid1 196 1  |-  ( F/ x ph  ->  ( E. x ph  <->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178   A.wal 1550   E.wex 1551   F/wnf 1554
This theorem is referenced by:  19.9h  1796  19.9d  1798  19.9OLD  1800  19.21t  1815  19.23t  1820  19.23tOLD  1840  spimt  1958  sbft  2119  vtoclegft  3029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-11 1763
This theorem depends on definitions:  df-bi 179  df-ex 1552  df-nf 1555
  Copyright terms: Public domain W3C validator