Theorem mtt 367

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 363	. 2 ⊢ (¬ 𝜑 → (¬ 𝜓 ↔ (¬ 𝜑 → ¬ 𝜓)))
2		con34b 319	. 2 ⊢ ((𝜓 → 𝜑) ↔ (¬ 𝜑 → ¬ 𝜓))
3	1, 2	bitr4di 292	1 ⊢ (¬ 𝜑 → (¬ 𝜓 ↔ (𝜓 → 𝜑)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209

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

This proof depends on definitions:  df-bi 210

This theorem is used by:  imnot  368  dfnot  1589  ralf0  4453  fnsuppres  8192  axpownd  10667  largei  32851  axpowg2  35788  axpowg3  35789