Theorem mt3i 150

Description: Modus tollens inference. (Contributed by NM, 26-Mar-1995.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)

Hypotheses

Ref	Expression
mt3i.1	⊢ ¬ 𝜒
mt3i.2	⊢ (𝜑 → (¬ 𝜓 → 𝜒))

Assertion

Ref	Expression
mt3i	⊢ (𝜑 → 𝜓)

Proof of Theorem mt3i

Step	Hyp	Ref	Expression
1		mt3i.1	. . 3 ⊢ ¬ 𝜒
2	1	a1i 11	. 2 ⊢ (𝜑 → ¬ 𝜒)
3		mt3i.2	. 2 ⊢ (𝜑 → (¬ 𝜓 → 𝜒))
4	2, 3	mt3d 149	1 ⊢ (𝜑 → 𝜓)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is used by:  ordeleqon  7787  wofib  9514  harcard  9980  infpssALT  10312  zorn2lem4  10498  lt6abl  20011  gzrngunitlem  21634  bwth  23619  i1f0rn  25894  lesrec  28045  dfon2lem3  36314  poimirlem30  38360