Theorem bj-currypara

Description: Curry's paradox. Note that the proof is intuitionistic (use of ax-3 comes from the unusual definition of the biconditional in set.mm). The paradox comes from the case where ph is the self-referential sentence "If this sentence is true, then ps ", 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 , such as affine logic and linear logic. (Contributed by BJ, 23-Sep-2023) (Proof modification is discouraged.)

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

Proof

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