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	⊢ ( 𝜓 ↔ ( Prv 𝜓 → 𝜑 ) )
	bj-babylob.1	⊢ ( Prv 𝜑 → 𝜑 )
Assertion	bj-babylob	⊢ 𝜑

Proof

Step	Hyp	Ref	Expression
1		bj-babylob.s	⊢ ( 𝜓 ↔ ( Prv 𝜓 → 𝜑 ) )
2		bj-babylob.1	⊢ ( Prv 𝜑 → 𝜑 )
3		ax-prv3	⊢ ( Prv 𝜓 → Prv Prv 𝜓 )
4	1	biimpi	⊢ ( 𝜓 → ( Prv 𝜓 → 𝜑 ) )
5	4	prvlem2	⊢ ( Prv 𝜓 → ( Prv Prv 𝜓 → Prv 𝜑 ) )
6	3 5	mpd	⊢ ( Prv 𝜓 → Prv 𝜑 )
7	6 2	syl	⊢ ( Prv 𝜓 → 𝜑 )
8	7 1	mpbir	⊢ 𝜓
9	8	ax-prv1	⊢ Prv 𝜓
10	9 7	ax-mp	⊢ 𝜑