Theorem dnege-P3.30 276

Description: Double Negation Elimination.

This statement is equivalent to ndexclmid-P3.16 181, thus it is not deducible with intuitionist logic.

Hypothesis

Ref	Expression
dnege-P3.30.1	⊢ (𝛾 → ¬ ¬ 𝜑)

Assertion

Ref	Expression
dnege-P3.30	⊢ (𝛾 → 𝜑)

Proof of Theorem dnege-P3.30

Step	Hyp	Ref	Expression
1		rcp-NDASM2of2 194	. 2 ⊢ ((𝛾 ∧ 𝜑) → 𝜑)
2		rcp-NDASM2of2 194	. . 3 ⊢ ((𝛾 ∧ ¬ 𝜑) → ¬ 𝜑)
3		dnege-P3.30.1	. . . 4 ⊢ (𝛾 → ¬ ¬ 𝜑)
4	3	rcp-NDIMP1add1 208	. . 3 ⊢ ((𝛾 ∧ ¬ 𝜑) → ¬ ¬ 𝜑)
5	2, 4	ndnege-P3.4 169	. 2 ⊢ ((𝛾 ∧ ¬ 𝜑) → 𝜑)
6		ndexclmid-P3.16.AC 251	. 2 ⊢ (𝛾 → (𝜑 ∨ ¬ 𝜑))
7	1, 5, 6	ndore-P3.12 177	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  df-or-D2.3 145  df-true-D2.4 155

This theorem is referenced by:  dnege-P3.30.RC  277  dnege-P3.30.CL  278  trnsp-P3.31b  282  trnsp-P3.31d  288  nmt-P3.32b  294  andasim-P3.46-L2  355  dmorgbfwd-L4.3  454  raa-P4  514  rcp-RAA2  516  rcp-RAA3  517  rcp-RAA4  518  rcp-RAA5  519  falseraa-P4  520  rcp-FALSERAA2-P  522  rcp-FALSERAA3  523  rcp-FALSERAA4  524  rcp-FALSERAA5  525