Theorem quantgodel 47412

Description: There can be no formula asserting its own non-universality, in parallel to bj-babygodel 37010; proof path is shorter but relying on a property of specialization which provability predicates do not have. For a matching proof, see quantgodelALT 47413. (Contributed by Ender Ting, 9-May-2026.)

Hypothesis

Ref	Expression
quantgodel.s	⊢ (𝜑 ↔ ¬ ∀𝑥𝜑)

Assertion

Ref	Expression
quantgodel	⊢ ⊥

Proof of Theorem quantgodel

Step	Hyp	Ref	Expression
1		sp 2217	. . . . . 6 ⊢ (∀𝑥𝜑 → 𝜑)
2		quantgodel.s	. . . . . 6 ⊢ (𝜑 ↔ ¬ ∀𝑥𝜑)
3	1, 2	sylib 220	. . . . 5 ⊢ (∀𝑥𝜑 → ¬ ∀𝑥𝜑)
4	3	pm2.01i 190	. . . 4 ⊢ ¬ ∀𝑥𝜑
5	4, 2	mpbir 233	. . 3 ⊢ 𝜑
6	5	ax-gen 1814	. 2 ⊢ ∀𝑥𝜑
7	6, 4	pm2.24ii 120	1 ⊢ ⊥

Syntax hints:  ¬ wn 3   ↔ wb 208  ∀wal 1557  ⊥wfal 1571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-12 2211

This theorem depends on definitions:  df-bi 209  df-ex 1799

This theorem is referenced by: (None)