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

Step	Hyp	Ref	Expression
1		id 22	. . 3 ⊢ (𝜑 → 𝜑)
2	1	exiftru 2060	. 2 ⊢ ∃𝑥(𝜑 → 𝜑)
3	2	19.35i 1958	1 ⊢ (∀𝑥𝜑 → ∃𝑥𝜑)

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