Theorem mtt 686

Description: Modus-tollens-like theorem. (Contributed by NM, 7-Apr-2001.) (Revised by Mario Carneiro, 31-Jan-2015.)

Assertion

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

Proof of Theorem mtt

Step	Hyp	Ref	Expression
1		pm2.21 618	. 2 ⊢ (¬ 𝜓 → (𝜓 → 𝜑))
2		con3 643	. . 3 ⊢ ((𝜓 → 𝜑) → (¬ 𝜑 → ¬ 𝜓))
3	2	com12 30	. 2 ⊢ (¬ 𝜑 → ((𝜓 → 𝜑) → ¬ 𝜓))
4	1, 3	impbid2 143	1 ⊢ (¬ 𝜑 → (¬ 𝜓 ↔ (𝜓 → 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  nbn2  698  dfnot  1382