Theorem mt2 191

Description: A rule similar to modus tollens. Inference associated with con2i 134. (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 185	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:  bijust  195  ax6dgen  2045  elirrv  8542  cardom  8850  0nnn  11090  nthruz  15026  hauspwdom  21352  fin1aufil  21783  rectbntr0  22682  lgam1  24835  gam1  24836  konigsberg  27235  ex-po  27422  strlem1  29237  eulerpartlemt  30561  nalf  32527  finxpreclem3  33360