Theorem bj-currypara

Description: Curry's paradox. Note that the proof is intuitionistic (use 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)

	Ref	Expression
Assertion	bj-currypara	|- ( ( ph <-> ( ph -> ps ) ) -> ps )

Proof

Step	Hyp	Ref	Expression
1		bj-animbi	|- ( ( ph /\ ps ) <-> ( ph <-> ( ph -> ps ) ) )
2		simpr	|- ( ( ph /\ ps ) -> ps )
3	1 2	sylbir	|- ( ( ph <-> ( ph -> ps ) ) -> ps )