Theorem bj-babylob 34713

Description: See the section header comments for the context, as well as the comments for bj-babygodel 34712.

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/ 34712).

(Contributed by BJ, 20-Apr-2019.)

Hypotheses

Ref	Expression
bj-babylob.s	⊢ (𝜓 ↔ (Prv 𝜓 → 𝜑))
bj-babylob.1	⊢ (Prv 𝜑 → 𝜑)

Assertion

Ref	Expression
bj-babylob	⊢ 𝜑

Proof of Theorem bj-babylob

Step	Hyp	Ref	Expression
1		ax-prv3 34709	. . . . . 6 ⊢ (Prv 𝜓 → Prv Prv 𝜓)
2		bj-babylob.s	. . . . . . . 8 ⊢ (𝜓 ↔ (Prv 𝜓 → 𝜑))
3	2	biimpi 215	. . . . . . 7 ⊢ (𝜓 → (Prv 𝜓 → 𝜑))
4	3	prvlem2 34711	. . . . . 6 ⊢ (Prv 𝜓 → (Prv Prv 𝜓 → Prv 𝜑))
5	1, 4	mpd 15	. . . . 5 ⊢ (Prv 𝜓 → Prv 𝜑)
6		bj-babylob.1	. . . . 5 ⊢ (Prv 𝜑 → 𝜑)
7	5, 6	syl 17	. . . 4 ⊢ (Prv 𝜓 → 𝜑)
8	7, 2	mpbir 230	. . 3 ⊢ 𝜓
9	8	ax-prv1 34707	. 2 ⊢ Prv 𝜓
10	9, 7	ax-mp 5	1 ⊢ 𝜑

Syntax hints:   → wi 4   ↔ wb 205  Prv cprvb 34706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-prv1 34707  ax-prv2 34708  ax-prv3 34709

This theorem depends on definitions:  df-bi 206

This theorem is referenced by:  bj-godellob  34714