Theorem 19.2 2011

Description: Theorem 19.2 of [Margaris] p. 89. This corresponds to the axiom (D) of modal logic (the other standard formulation being extru 2010). 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 2228 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 1817). (Contributed by NM, 2-Aug-2017.) Remove dependency on ax-7 2043. (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 23	. . 3 ⊢ (𝜑 → 𝜑)
2	1	exgen 2009	. 2 ⊢ ∃𝑥(𝜑 → 𝜑)
3	2	19.35i 1913	1 ⊢ (∀𝑥𝜑 → ∃𝑥𝜑)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-6 2002

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by:  19.2d  2012  19.39  2025  19.24  2026  19.34  2027  eusv2i  5363  bj-ax6e  37782  bj-spnfw  37785  bj-modald  37788  wl-speqv  38722  wl-19.8eqv  38723  pm10.251  45605  ax6e2eq  45800  ax6e2eqVD  46149