Theorem mt2 202

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 195	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  206  ax6dgen  2161  nlim2  8453  infn0  9240  elirrvOLDOLD  9541  cardom  9938  0nnnALT  12244  nthruz  16276  hauspwdom  23549  fin1aufil  23980  rectbntr0  24881  lgam1  27116  gam1  27117  konigsberg  30416  ex-po  30594  strlem1  32410  eulerpartlemt  34629  nalfal  36724  bj-mt2bi  36971  finxpreclem3  37848  nregmodel  45554  quantgodelALT  47410  tannpoly  47445