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

Theorem nfdi 2019
Description: Since the converse holds by a1i 11, this inference shows that we can represent a not-free hypothesis with either  F/ x ph (inference form) or  ( ph  ->  F/ x ph ) (deduction form). (Contributed by NM, 17-Aug-2018.) (Proof shortened by Wolf Lammen, 10-Jul-2019.)
Hypothesis
Ref Expression
nfdi.1  |-  ( ph  ->  F/ x ph )
Assertion
Ref Expression
nfdi  |-  F/ x ph

Proof of Theorem nfdi
StepHypRef Expression
1 nfdi.1 . . . 4  |-  ( ph  ->  F/ x ph )
21nfrd 1973 . . 3  |-  ( ph  ->  ( ph  ->  A. x ph ) )
32pm2.43i 48 . 2  |-  ( ph  ->  A. x ph )
43nfi 1682 1  |-  F/ x ph
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1450   F/wnf 1675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-12 1950
This theorem depends on definitions:  df-bi 190  df-ex 1672  df-nf 1676
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator