Theorem mt2bi 328

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 327	. 2 ⊢ (¬ ψ ↔ (φ → ¬ ψ))
3		con2b 324	. 2 ⊢ ((φ → ¬ ψ) ↔ (ψ → ¬ φ))
4	2, 3	bitri 240	1 ⊢ (¬ ψ ↔ (ψ → ¬ φ))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 176

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177

This theorem is used by: (None)