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

Theorem 19.2 2080
Description: Theorem 19.2 of [Margaris] p. 89. This corresponds to the axiom (D) of modal logic (the other standard formulation being extru 2079). 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 2229 for a more conventional proof of a more general result, which uses additional axioms. The reverse implication is the defining property of effective nonfreeness (see df-nf 1883). (Contributed by NM, 2-Aug-2017.) Remove dependency on ax-7 2112. (Revised by Wolf Lammen, 4-Dec-2017.)
Assertion
Ref Expression
19.2 (∀𝑥𝜑 → ∃𝑥𝜑)

Proof of Theorem 19.2
StepHypRef Expression
1 id 22 . . 3 (𝜑𝜑)
21exgen 2078 . 2 𝑥(𝜑𝜑)
3219.35i 1981 1 (∀𝑥𝜑 → ∃𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1654  wex 1878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-6 2075
This theorem depends on definitions:  df-bi 199  df-ex 1879
This theorem is referenced by:  19.2d  2081  19.39  2087  19.24  2088  19.34  2089  eusv2i  5096  bj-ax6e  33188  bj-spnfw  33192  bj-modald  33195  wl-speqv  33852  wl-19.8eqv  33853  pm10.251  39398  ax6e2eq  39600  ax6e2eqVD  39960
  Copyright terms: Public domain W3C validator