Theorem currybi 34657

Description: Biconditional version of Curry's paradox. If some proposition 𝜑 amounts to the self-referential statement "This very statement is equivalent to 𝜓", then 𝜓 is true. See bj-currypara 35424 in BJ's mathbox for the classical version. (Contributed by Adrian Ducourtial, 18-Mar-2025.)

Assertion

Ref	Expression
currybi	⊢ ((𝜑 ↔ (𝜑 ↔ 𝜓)) → 𝜓)

Proof of Theorem currybi

Step	Hyp	Ref	Expression
1		biid 260	. 2 ⊢ (𝜑 ↔ 𝜑)
2		biass 385	. . 3 ⊢ (((𝜑 ↔ 𝜑) ↔ 𝜓) ↔ (𝜑 ↔ (𝜑 ↔ 𝜓)))
3	2	biimpri 227	. 2 ⊢ ((𝜑 ↔ (𝜑 ↔ 𝜓)) → ((𝜑 ↔ 𝜑) ↔ 𝜓))
4	1, 3	mpbii 232	1 ⊢ ((𝜑 ↔ (𝜑 ↔ 𝜓)) → 𝜓)

Syntax hints:   → wi 4   ↔ wb 205

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 206

This theorem is referenced by: (None)