Theorem mt2 199

Description: A rule similar to modus tollens. Inference associated with con2i 139. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 10-Sep-2013.)

Hypotheses

Ref	Expression
mt2.1	⊢ 𝜓
mt2.2	⊢ (𝜑 → ¬ 𝜓)

Assertion

Ref	Expression
mt2	⊢ ¬ 𝜑

Proof of Theorem mt2

Step	Hyp	Ref	Expression
1		mt2.1	. . 3 ⊢ 𝜓
2	1	a1i 11	. 2 ⊢ (𝜑 → 𝜓)
3		mt2.2	. 2 ⊢ (𝜑 → ¬ 𝜓)
4	2, 3	pm2.65i 193	1 ⊢ ¬ 𝜑

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:  bijust0  203  ax6dgen  2126  elirrv  9285  cardom  9675  0nnnALT  11940  nthruz  15890  hauspwdom  22560  fin1aufil  22991  rectbntr0  23901  lgam1  26118  gam1  26119  konigsberg  28522  ex-po  28700  strlem1  30513  eulerpartlemt  32238  nalfal  34519  bj-mt2bi  34676  finxpreclem3  35491