Theorem bj-currypara 36561

Description: Curry's paradox. Note that the proof is intuitionistic (use of ax-3 8 comes from the unusual definition of the biconditional in set.mm). The paradox comes from the case where 𝜑 is the self-referential sentence "If this sentence is true, then 𝜓", so that one can prove everything. Therefore, a consistent system cannot allow the formation of such self-referential sentences. This has lead to the study of logics rejecting contraction pm2.43 56, such as affine logic and linear logic. (Contributed by BJ, 23-Sep-2023.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-currypara	⊢ ((𝜑 ↔ (𝜑 → 𝜓)) → 𝜓)

Proof of Theorem bj-currypara

Step	Hyp	Ref	Expression
1		bj-animbi 36560	. 2 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜑 ↔ (𝜑 → 𝜓)))
2		simpr 484	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜓)
3	1, 2	sylbir 235	1 ⊢ ((𝜑 ↔ (𝜑 → 𝜓)) → 𝜓)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 207  df-an 396

This theorem is referenced by: (None)