Theorem 19.2 1892

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 2058 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 1710). (Contributed by NM, 2-Aug-2017.) Remove dependency on ax-7 1935. (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 1891	. 2 ⊢ ∃𝑥(𝜑 → 𝜑)
3	2	19.35i 1806	1 ⊢ (∀𝑥𝜑 → ∃𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1481  ∃wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-6 1888

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by:  19.2d  1893  19.39  1899  19.24  1900  19.34  1901  eusv2i  4863  extt  32403  bj-ax6e  32653  bj-spnfw  32658  bj-modald  32661  wl-speqv  33308  wl-19.8eqv  33309  pm10.251  38559  ax6e2eq  38773  ax6e2eqVD  39143