Theorem mtt 329

Description: Modus-tollens-like theorem. (Contributed by NM, 7-Apr-2001.) (Proof shortened by Wolf Lammen, 12-Nov-2012.)

Assertion

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

Proof of Theorem mtt

Step	Hyp	Ref	Expression
1		biimt 325	. 2 ⊢ (¬ φ → (¬ ψ ↔ (¬ φ → ¬ ψ)))
2		con34b 283	. 2 ⊢ ((ψ → φ) ↔ (¬ φ → ¬ ψ))
3	1, 2	syl6bbr 254	1 ⊢ (¬ φ → (¬ 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)