Theorem mtt 354

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 350	. 2 ⊢ (¬ 𝜑 → (¬ 𝜓 ↔ (¬ 𝜑 → ¬ 𝜓)))
2		con34b 306	. 2 ⊢ ((𝜓 → 𝜑) ↔ (¬ 𝜑 → ¬ 𝜓))
3	1, 2	syl6bbr 278	1 ⊢ (¬ 𝜑 → (¬ 𝜓 ↔ (𝜓 → 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196

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 197

This theorem is referenced by:  imnot  355  dfnot  1502  ralf0  4078  fnsuppres  7322  axpownd  9423  largei  29126