Theorem mtoi 169

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 10	. 2 ⊢ (φ → ¬ χ)
3		mtoi.2	. 2 ⊢ (φ → (ψ → χ))
4	2, 3	mtod 168	1 ⊢ (φ → ¬ ψ)

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is referenced by:  mtbii  293  mtbiri  294  sp  1747  spOLD  1748  equsalhwOLD  1839  dvelimv  1939  ax9o  1950  nndisjeq  4430  sfinltfin  4536  vfin1cltv  4548  nulnnn  4557  nmembers1lem2  6270