Theorem bj-currypara

Description: Curry's paradox. 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	$$⊢\left(\phi \leftrightarrow \left(\phi \to \psi \right)\right)\to \psi$$

Proof

Step	Hyp	Ref	Expression
1		bj-animbi	$$⊢\left(\phi \wedge \psi \right)\leftrightarrow \left(\phi \leftrightarrow \left(\phi \to \psi \right)\right)$$
2		simpr	$$⊢\left(\phi \wedge \psi \right)\to \psi$$
3	1 2	sylbir	$$⊢\left(\phi \leftrightarrow \left(\phi \to \psi \right)\right)\to \psi$$