Theorem bj-godellob 34714

Description: Proof of Gödel's theorem from Löb's theorem (see comments at bj-babygodel 34712 and bj-babylob 34713 for details). (Contributed by BJ, 20-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bj-godellob.s	⊢ (𝜑 ↔ ¬ Prv 𝜑)
bj-godellob.1	⊢ ¬ Prv ⊥

Assertion

Ref	Expression
bj-godellob	⊢ ⊥

Proof of Theorem bj-godellob

Step	Hyp	Ref	Expression
1		bj-godellob.s	. . 3 ⊢ (𝜑 ↔ ¬ Prv 𝜑)
2		dfnot 1558	. . 3 ⊢ (¬ Prv 𝜑 ↔ (Prv 𝜑 → ⊥))
3	1, 2	bitri 274	. 2 ⊢ (𝜑 ↔ (Prv 𝜑 → ⊥))
4		bj-godellob.1	. . 3 ⊢ ¬ Prv ⊥
5	4	pm2.21i 119	. 2 ⊢ (Prv ⊥ → ⊥)
6	3, 5	bj-babylob 34713	1 ⊢ ⊥

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205  ⊥wfal 1551  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  df-tru 1542  df-fal 1552

This theorem is referenced by: (None)