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Theorem 19.2 1981
Description: Theorem 19.2 of [Margaris] p. 89. This corresponds to the axiom (D) of modal logic (the other standard formulation being extru 1980). 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 2187 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 1785). (Contributed by NM, 2-Aug-2017.) Remove dependency on ax-7 2015. (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 1978 . 2 𝑥(𝜑𝜑)
3219.35i 1879 1 (∀𝑥𝜑 → ∃𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1535  wex 1780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-6 1970
This theorem depends on definitions:  df-bi 209  df-ex 1781
This theorem is referenced by:  19.2d  1982  19.39  1991  19.24  1992  19.34  1993  eusv2i  5271  bj-ax6e  34009  bj-spnfw  34011  bj-modald  34014  wl-speqv  34803  wl-19.8eqv  34804  pm10.251  40847  ax6e2eq  41046  ax6e2eqVD  41396
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