Theorem mt2bi 641

Description: A false consequent falsifies an antecedent. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Nov-2012.)

Hypothesis

Ref	Expression
mt2bi.1	⊢ 𝜑

Assertion

Ref	Expression
mt2bi	⊢ (¬ 𝜓 ↔ (𝜓 → ¬ 𝜑))

Proof of Theorem mt2bi

Step	Hyp	Ref	Expression
1		mt2bi.1	. . 3 ⊢ 𝜑
2	1	a1bi 241	. 2 ⊢ (¬ 𝜓 ↔ (𝜑 → ¬ 𝜓))
3		con2b 625	. 2 ⊢ ((𝜑 → ¬ 𝜓) ↔ (𝜓 → ¬ 𝜑))
4	2, 3	bitri 182	1 ⊢ (¬ 𝜓 ↔ (𝜓 → ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577

This theorem depends on definitions:  df-bi 115

This theorem is referenced by: (None)