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	|- ( ps <-> ( Prv ps -> ph ) )
	bj-babylob.1	|- ( Prv ph -> ph )
Assertion	bj-babylob	|- ph

Proof

Step	Hyp	Ref	Expression
1		bj-babylob.s	|- ( ps <-> ( Prv ps -> ph ) )
2		bj-babylob.1	|- ( Prv ph -> ph )
3		ax-prv3	|- ( Prv ps -> Prv Prv ps )
4	1	biimpi	|- ( ps -> ( Prv ps -> ph ) )
5	4	prvlem2	|- ( Prv ps -> ( Prv Prv ps -> Prv ph ) )
6	3 5	mpd	|- ( Prv ps -> Prv ph )
7	6 2	syl	|- ( Prv ps -> ph )
8	7 1	mpbir	|- ps
9	8	ax-prv1	|- Prv ps
10	9 7	ax-mp	|- ph