Theorem bj-currypeirce 32544

Description: Curry's axiom (a non-intuitionistic statement sometimes called a paradox of material implication) implies Peirce's axiom peirce 193 over minimal implicational calculus and the axiomatic definition of disjunction (olc 399, orc 400, jao 534). A shorter proof from bj-orim2 32541, pm1.2 535, syl6com 37 is possible if we accept to use pm1.2 535, itself a direct consequence of jao 534. (Contributed by BJ, 15-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-currypeirce	⊢ ((𝜑 ∨ (𝜑 → 𝜓)) → (((𝜑 → 𝜓) → 𝜑) → 𝜑))

Proof of Theorem bj-currypeirce

Step	Hyp	Ref	Expression
1		olc 399	. . 3 ⊢ (𝜑 → (𝜑 ∨ 𝜑))
2	1	imim2i 16	. . 3 ⊢ (((𝜑 → 𝜓) → 𝜑) → ((𝜑 → 𝜓) → (𝜑 ∨ 𝜑)))
3		jao 534	. . 3 ⊢ ((𝜑 → (𝜑 ∨ 𝜑)) → (((𝜑 → 𝜓) → (𝜑 ∨ 𝜑)) → ((𝜑 ∨ (𝜑 → 𝜓)) → (𝜑 ∨ 𝜑))))
4	1, 2, 3	mpsyl 68	. 2 ⊢ (((𝜑 → 𝜓) → 𝜑) → ((𝜑 ∨ (𝜑 → 𝜓)) → (𝜑 ∨ 𝜑)))
5		id 22	. . 3 ⊢ (𝜑 → 𝜑)
6		jao 534	. . 3 ⊢ ((𝜑 → 𝜑) → ((𝜑 → 𝜑) → ((𝜑 ∨ 𝜑) → 𝜑)))
7	5, 5, 6	mp2 9	. 2 ⊢ ((𝜑 ∨ 𝜑) → 𝜑)
8	4, 7	syl6com 37	1 ⊢ ((𝜑 ∨ (𝜑 → 𝜓)) → (((𝜑 → 𝜓) → 𝜑) → 𝜑))

Syntax hints:   → wi 4   ∨ wo 383

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 197  df-or 385  df-an 386

This theorem is referenced by: (None)