MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  19.2 Structured version   Visualization version   Unicode version

Theorem 19.2 1820
Description: Theorem 19.2 of [Margaris] p. 89. This corresponds to the axiom (D) of modal logic. Note: This proof is very different from Margaris' because we only have Tarski's FOL axiom schemes available at this point. See the later 19.2g 1957 for a more conventional proof of a more general result, which uses additional axioms. (Contributed by NM, 2-Aug-2017.) Remove dependency on ax-7 1862. (Revised by Wolf Lammen, 4-Dec-2017.)
Assertion
Ref Expression
19.2  |-  ( A. x ph  ->  E. x ph )

Proof of Theorem 19.2
StepHypRef Expression
1 id 22 . . 3  |-  ( ph  ->  ph )
21exiftru 1819 . 2  |-  E. x
( ph  ->  ph )
3219.35i 1752 1  |-  ( A. x ph  ->  E. x ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1453   E.wex 1674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-6 1816
This theorem depends on definitions:  df-bi 190  df-ex 1675
This theorem is referenced by:  19.8w  1821  19.39  1826  19.24  1827  19.34  1828  eusv2i  4614  extt  31113  bj-ax6e  31311  bj-19.8w  31313  bj-spnfw  31315  pm10.251  36753  ax6e2eq  36968  ax6e2eqVD  37344
  Copyright terms: Public domain W3C validator