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

Step	Hyp	Ref	Expression
1		id 22	. . 3 ⊢ (𝜑 → 𝜑)
2	1	exgen 1978	. 2 ⊢ ∃𝑥(𝜑 → 𝜑)
3	2	19.35i 1879	1 ⊢ (∀𝑥𝜑 → ∃𝑥𝜑)

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  5295  bj-ax6e  34001  bj-spnfw  34003  bj-modald  34006  wl-speqv  34777  wl-19.8eqv  34778  pm10.251  40712  ax6e2eq  40911  ax6e2eqVD  41261