Theorem mtoi 577

Description: Modus tollens inference. (Contributed by NM, 5-Jul-1994.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)

Hypotheses

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

Assertion

Ref	Expression
mtoi	⊢ (φ → ¬ ψ)

Proof of Theorem mtoi

Step	Hyp	Ref	Expression
1		mtoi.1	. . 3 ⊢ ¬ χ
2	1	a1i 9	. 2 ⊢ (φ → ¬ χ)
3		mtoi.2	. 2 ⊢ (φ → (ψ → χ))
4	2, 3	mtod 576	1 ⊢ (φ → ¬ ψ)

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-in1 532  ax-in2 533

This theorem is referenced by:  mtbii  586  mtbiri  587  pwnss  3886  nsuceq0g  4104  ordsuc  4225  onnmin  4228  ssnel  4229  onpsssuc  4231  acexmidlemab  5430  reldmtpos  5790  dmtpos  5793  2pwuninelg  5820  ltexprlemdisj  6443  recexprlemdisj  6464