Theorem 19.2 1879

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 2046 for a more conventional proof of a more general result, which uses additional axioms. (Contributed by NM, 2-Aug-2017.) Remove dependency on ax-7 1922. (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 1878	. 2 ⊢ ∃𝑥(𝜑 → 𝜑)
3	2	19.35i 1795	1 ⊢ (∀𝑥𝜑 → ∃𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1473  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-6 1875

This theorem depends on definitions:  df-bi 196  df-ex 1696

This theorem is referenced by:  19.2d  1880  19.39  1886  19.24  1887  19.34  1888  eusv2i  4789  extt  31573  bj-ax6e  31842  bj-spnfw  31845  bj-modald  31848  wl-speqv  32487  wl-19.8eqv  32488  pm10.251  37581  ax6e2eq  37794  ax6e2eqVD  38165