Theorem mtod 168

Description: Modus tollens deduction. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)

Hypotheses

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

Assertion

Ref	Expression
mtod	⊢ (φ → ¬ ψ)

Proof of Theorem mtod

Step	Hyp	Ref	Expression
1		mtod.2	. 2 ⊢ (φ → (ψ → χ))
2		mtod.1	. . 3 ⊢ (φ → ¬ χ)
3	2	a1d 22	. 2 ⊢ (φ → (ψ → ¬ χ))
4	1, 3	pm2.65d 166	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:  mtoi  169  mtbid  291  mtbird  292  mtand  640  mtord  641  ltfinasym  4461  lenltfin  4470  tfinltfin  4502  nnadjoin  4521  sfin111  4537  vfinncvntnn  4549  enprmaplem3  6079