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Theorem bj-babylob 32258
Description: See the section header comments for the context, as well as the comments for bj-babygodel 32257.

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 (𝜓 ↔ (Prv 𝜓𝜑))
bj-babylob.1 (Prv 𝜑𝜑)
Assertion
Ref Expression
bj-babylob 𝜑

Proof of Theorem bj-babylob
StepHypRef Expression
1 ax-prv3 32254 . . . . . 6 (Prv 𝜓 → Prv Prv 𝜓)
2 bj-babylob.s . . . . . . . 8 (𝜓 ↔ (Prv 𝜓𝜑))
32biimpi 206 . . . . . . 7 (𝜓 → (Prv 𝜓𝜑))
43prvlem2 32256 . . . . . 6 (Prv 𝜓 → (Prv Prv 𝜓 → Prv 𝜑))
51, 4mpd 15 . . . . 5 (Prv 𝜓 → Prv 𝜑)
6 bj-babylob.1 . . . . 5 (Prv 𝜑𝜑)
75, 6syl 17 . . . 4 (Prv 𝜓𝜑)
87, 2mpbir 221 . . 3 𝜓
98ax-prv1 32252 . 2 Prv 𝜓
109, 7ax-mp 5 1 𝜑
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  Prv cprvb 32251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-prv1 32252  ax-prv2 32253  ax-prv3 32254
This theorem depends on definitions:  df-bi 197
This theorem is referenced by:  bj-godellob  32259
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