Theorem bicontr 38071

Description: Biconditional of its own negation is a contradiction. (Contributed by Giovanni Mascellani, 15-Sep-2017.)

Assertion

Ref	Expression
bicontr	⊢ ((¬ 𝜑 ↔ 𝜑) ↔ ⊥)

Proof of Theorem bicontr

Step	Hyp	Ref	Expression
1		biid 261	. . 3 ⊢ (𝜑 ↔ 𝜑)
2		notbinot1 38070	. . 3 ⊢ (¬ (¬ 𝜑 ↔ 𝜑) ↔ (𝜑 ↔ 𝜑))
3	1, 2	mpbir 231	. 2 ⊢ ¬ (¬ 𝜑 ↔ 𝜑)
4	3	bifal 1556	1 ⊢ ((¬ 𝜑 ↔ 𝜑) ↔ ⊥)

Syntax hints:  ¬ wn 3   ↔ wb 206  ⊥wfal 1552

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 207  df-tru 1543  df-fal 1553

This theorem is referenced by: (None)