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Theorem 19.2 2061
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 2212 for a more conventional proof of a more general result, which uses additional axioms. The reverse implication is the defining property of effective non-freeness (see df-nf 1858). (Contributed by NM, 2-Aug-2017.) Remove dependency on ax-7 2093. (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)
21exiftru 2060 . 2 𝑥(𝜑𝜑)
3219.35i 1958 1 (∀𝑥𝜑 → ∃𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1629  wex 1852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-6 2057
This theorem depends on definitions:  df-bi 197  df-ex 1853
This theorem is referenced by:  19.2d  2062  19.39  2068  19.24  2069  19.34  2070  eusv2i  4994  extt  32740  bj-ax6e  32990  bj-spnfw  32995  bj-modald  32998  wl-speqv  33644  wl-19.8eqv  33645  pm10.251  39085  ax6e2eq  39298  ax6e2eqVD  39665
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