Theorem dnegeint-P4.12 421

Description: Version of Double Negative Elimination deducible with intuitionist logic. †

Hypothesis

Ref	Expression
dnegeint-P4.12.1	⊢ (𝛾 → ¬ ¬ ¬ 𝜑)

Assertion

Ref	Expression
dnegeint-P4.12	⊢ (𝛾 → ¬ 𝜑)

Proof of Theorem dnegeint-P4.12

Step	Hyp	Ref	Expression
1		rcp-NDASM2of2 194	. . 3 ⊢ ((𝛾 ∧ 𝜑) → 𝜑)
2	1	dnegi-P3.29 273	. 2 ⊢ ((𝛾 ∧ 𝜑) → ¬ ¬ 𝜑)
3		dnegeint-P4.12.1	. . 3 ⊢ (𝛾 → ¬ ¬ ¬ 𝜑)
4	3	rcp-NDIMP1add1 208	. 2 ⊢ ((𝛾 ∧ 𝜑) → ¬ ¬ ¬ 𝜑)
5	2, 4	rcp-NDNEGI2 219	1 ⊢ (𝛾 → ¬ 𝜑)

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

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

This theorem is referenced by:  dnegeint-P4.12.RC  422  dnegeqint-P4.13  423