Theorem exclmid2peirce-P4.41a 512

Description: Peirce's Law from Law of Excluded Middle. †

The use of ndexclmid-P3.16 181 is simply replaced with a hypothesis. All the other rules used are valid in intuitionist logic.

Hypothesis

Ref	Expression
exclmid2peirce-P4.41a.1	⊢ (𝜑 ∨ ¬ 𝜑)

Assertion

Ref	Expression
exclmid2peirce-P4.41a	⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜑)

Proof of Theorem exclmid2peirce-P4.41a

Step	Hyp	Ref	Expression
1		rcp-NDASM2of2 194	. 2 ⊢ ((((𝜑 → 𝜓) → 𝜑) ∧ 𝜑) → 𝜑)
2		rcp-NDASM2of2 194	. . . 4 ⊢ ((((𝜑 → 𝜓) → 𝜑) ∧ ¬ 𝜑) → ¬ 𝜑)
3	2	impoe-P4.4a 377	. . 3 ⊢ ((((𝜑 → 𝜓) → 𝜑) ∧ ¬ 𝜑) → (𝜑 → 𝜓))
4		rcp-NDASM1of2 193	. . 3 ⊢ ((((𝜑 → 𝜓) → 𝜑) ∧ ¬ 𝜑) → ((𝜑 → 𝜓) → 𝜑))
5	3, 4	ndime-P3.6 171	. 2 ⊢ ((((𝜑 → 𝜓) → 𝜑) ∧ ¬ 𝜑) → 𝜑)
6		exclmid2peirce-P4.41a.1	. . 3 ⊢ (𝜑 ∨ ¬ 𝜑)
7	6	rcp-NDIMP0addall 207	. 2 ⊢ (((𝜑 → 𝜓) → 𝜑) → (𝜑 ∨ ¬ 𝜑))
8	1, 5, 7	rcp-NDORE2 235	1 ⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜑)

Syntax hints:  ¬ wff-neg 9   → wff-imp 10   ∧ wff-and 132   ∨ wff-or 144

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-or-D2.3 145  df-true-D2.4 155

This theorem is referenced by: (None)