Theorem bj-babylob

Description: See the section header comments for the context, as well as the comments for bj-babygodel .

Löb's theorem when the Löb sentence is given as a hypothesis (the hard part of the proof of Löb's theorem is to construct this Löb sentence; this can be done, using Gödel diagonalization, for any first-order effectively axiomatizable theory containing Robinson arithmetic). More precisely, the present theorem states that if a first-order theory proves that the provability of a given sentence entails its truth (and if one can construct in this theory a provability predicate and a Löb sentence, given here as the first hypothesis), then the theory actually proves that sentence.

See for instance, Eliezer Yudkowsky,The Cartoon Guide to Löb's Theorem (available at http://yudkowsky.net/rational/lobs-theorem/ ).

(Contributed by BJ, 20-Apr-2019)

	Ref	Expression
Hypotheses	bj-babylob.s	$$⊢\psi \leftrightarrow \left(Prv\psi \to \phi \right)$$
	bj-babylob.1	$$⊢Prv\phi \to \phi$$
Assertion	bj-babylob	$$⊢\phi$$

Proof

Step	Hyp	Ref	Expression
1		bj-babylob.s	$$⊢\psi \leftrightarrow \left(Prv\psi \to \phi \right)$$
2		bj-babylob.1	$$⊢Prv\phi \to \phi$$
3		ax-prv3	$$⊢Prv\psi \to PrvPrv\psi$$
4	1	biimpi	$$⊢\psi \to \left(Prv\psi \to \phi \right)$$
5	4	prvlem2	$$⊢Prv\psi \to \left(PrvPrv\psi \to Prv\phi \right)$$
6	3 5	mpd	$$⊢Prv\psi \to Prv\phi$$
7	6 2	syl	$$⊢Prv\psi \to \phi$$
8	7 1	mpbir	$$⊢\psi$$
9	8	ax-prv1	$$⊢Prv\psi$$
10	9 7	ax-mp	$$⊢\phi$$