Theorem 19.2 2080

Description: Theorem 19.2 of [Margaris] p. 89. This corresponds to the axiom (D) of modal logic (the other standard formulation being extru 2079). 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 2229 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 1883). (Contributed by NM, 2-Aug-2017.) Remove dependency on ax-7 2112. (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 2078	. 2 ⊢ ∃𝑥(𝜑 → 𝜑)
3	2	19.35i 1981	1 ⊢ (∀𝑥𝜑 → ∃𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1654  ∃wex 1878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-6 2075

This theorem depends on definitions:  df-bi 199  df-ex 1879

This theorem is referenced by:  19.2d  2081  19.39  2087  19.24  2088  19.34  2089  eusv2i  5096  bj-ax6e  33188  bj-spnfw  33192  bj-modald  33195  wl-speqv  33852  wl-19.8eqv  33853  pm10.251  39398  ax6e2eq  39600  ax6e2eqVD  39960