Theorem bj-peirce 33068

Description: Proof of peirce 194 from minimal implicational calculus, the axiomatic definition of disjunction (olc 899, orc 898, jao 988), and Curry's axiom curryax 922. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-peirce	⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜑)

Proof of Theorem bj-peirce

Step	Hyp	Ref	Expression
1		curryax 922	. . 3 ⊢ (𝜑 ∨ (𝜑 → 𝜓))
2		bj-orim2 33067	. . 3 ⊢ (((𝜑 → 𝜓) → 𝜑) → ((𝜑 ∨ (𝜑 → 𝜓)) → (𝜑 ∨ 𝜑)))
3	1, 2	mpi 20	. 2 ⊢ (((𝜑 → 𝜓) → 𝜑) → (𝜑 ∨ 𝜑))
4		pm1.2 932	. 2 ⊢ ((𝜑 ∨ 𝜑) → 𝜑)
5	3, 4	syl 17	1 ⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜑)

Syntax hints:   → wi 4   ∨ wo 878

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 199  df-an 387  df-or 879

This theorem is referenced by: (None)