Theorem pm4.81 394

Description: A formula is equivalent to its negation implying it. Theorem *4.81 of [WhiteheadRussell] p. 122. Note that the second step, using pm2.24 124, could also use ax-1 6. (Contributed by NM, 3-Jan-2005.)

Assertion

Ref	Expression
pm4.81	⊢ ((¬ 𝜑 → 𝜑) ↔ 𝜑)

Proof of Theorem pm4.81

Step	Hyp	Ref	Expression
1		pm2.18 128	. 2 ⊢ ((¬ 𝜑 → 𝜑) → 𝜑)
2		pm2.24 124	. 2 ⊢ (𝜑 → (¬ 𝜑 → 𝜑))
3	1, 2	impbii 208	1 ⊢ ((¬ 𝜑 → 𝜑) ↔ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205

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 206

This theorem is referenced by:  ifpimimb  41111  ifpimim  41116