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

Theorem 19.9t 1944
Description: A closed version of 19.9 1945. (Contributed by NM, 13-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) (Proof shortened by Wolf Lammen, 14-Jul-2020.)
Assertion
Ref Expression
19.9t  |-  ( F/ x ph  ->  ( E. x ph  <->  ph ) )

Proof of Theorem 19.9t
StepHypRef Expression
1 id 23 . . 3  |-  ( F/ x ph  ->  F/ x ph )
2119.9d 1943 . 2  |-  ( F/ x ph  ->  ( E. x ph  ->  ph )
)
3 19.8a 1910 . 2  |-  ( ph  ->  E. x ph )
42, 3impbid1 206 1  |-  ( F/ x ph  ->  ( E. x ph  <->  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187   E.wex 1659   F/wnf 1663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-12 1907
This theorem depends on definitions:  df-bi 188  df-ex 1660  df-nf 1664
This theorem is referenced by:  19.9  1945  19.9hOLD  1948  19.9dOLD  1949  19.21t  1961  19.23tOLD  1967  spimt  2061  sbft  2174  vtoclegft  3159  bj-cbv3tb  31059  bj-spimtv  31066  bj-sbftv  31133  bj-equsal1t  31183  bj-19.21t  31191
  Copyright terms: Public domain W3C validator