Theorem mt2i 138

Description: Modus tollens inference. (Contributed by NM, 26-Mar-1995.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)

Hypotheses

Ref	Expression
mt2i.1	⊢ 𝜒
mt2i.2	⊢ (𝜑 → (𝜓 → ¬ 𝜒))

Assertion

Ref	Expression
mt2i	⊢ (𝜑 → ¬ 𝜓)

Proof of Theorem mt2i

Step	Hyp	Ref	Expression
1		mt2i.1	. . 3 ⊢ 𝜒
2	1	a1i 11	. 2 ⊢ (𝜑 → 𝜒)
3		mt2i.2	. 2 ⊢ (𝜑 → (𝜓 → ¬ 𝜒))
4	2, 3	mt2d 137	1 ⊢ (𝜑 → ¬ 𝜓)

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

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

This theorem is used by:  ssnlim  7884  elirrvOLDOLD  9564  konigthlem  10564  ipo0  45191  ifr0  45192  gpg5nbgrvtx03star  48878  gpg5nbgr3star  48879