Theorem mt2 192

Description: A rule similar to modus tollens. Inference associated with con2i 137. (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 186	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  196  ax6dgen  2179  elirrv  8777  cardom  9132  0nnnALT  11395  nthruz  15363  hauspwdom  21682  fin1aufil  22113  rectbntr0  23012  lgam1  25210  gam1  25211  konigsberg  27632  ex-po  27846  strlem1  29660  eulerpartlemt  30974  nalfal  32931  finxpreclem3  33770