Theorem bj-babylob 32589

Description: See the section header comments for the context, as well as the comments for bj-babygodel 32588.

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.)

Hypotheses

Ref	Expression
bj-babylob.s	|- ( ps <-> (Prv ps -> ph ) )
bj-babylob.1	|- (Prv ph -> ph )

Assertion

Ref	Expression
bj-babylob	|- ph

Proof of Theorem bj-babylob

Step	Hyp	Ref	Expression
1		ax-prv3 32585	. . . . . 6 |- (Prv ps -> Prv Prv ps )
2		bj-babylob.s	. . . . . . . 8 |- ( ps <-> (Prv ps -> ph ) )
3	2	biimpi 206	. . . . . . 7 |- ( ps -> (Prv ps -> ph ) )
4	3	prvlem2 32587	. . . . . 6 |- (Prv ps -> (Prv Prv ps -> Prv ph ) )
5	1, 4	mpd 15	. . . . 5 |- (Prv ps -> Prv ph )
6		bj-babylob.1	. . . . 5 |- (Prv ph -> ph )
7	5, 6	syl 17	. . . 4 |- (Prv ps -> ph )
8	7, 2	mpbir 221	. . 3 |- ps
9	8	ax-prv1 32583	. 2 |- Prv ps
10	9, 7	ax-mp 5	1 |- ph

Syntax hints:   -> wi 4   <-> wb 196  Prv cprvb 32582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-prv1 32583  ax-prv2 32584  ax-prv3 32585

This theorem depends on definitions:  df-bi 197

This theorem is referenced by:  bj-godellob  32590