Theorem mt2 190

Description: A rule similar to modus tollens. Inference associated with con2i 133. (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 184	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  194  ax6dgen  1992  elirrv  8387  cardom  8695  0nnn  10929  nthruz  14820  hauspwdom  21114  fin1aufil  21546  rectbntr0  22443  lgam1  24590  gam1  24591  wlkntrl  26092  ex-po  26684  strlem1  28493  eulerpartlemt  29760  nalf  31572  finxpreclem3  32406  konigsberg-av  41427