Theorem nsyli 156

Description: A negated syllogism inference. (Contributed by NM, 3-May-1994.)

Hypotheses

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

Assertion

Ref	Expression
nsyli	⊢ (𝜑 → (𝜃 → ¬ 𝜓))

Proof of Theorem nsyli

Step	Hyp	Ref	Expression
1		nsyli.2	. 2 ⊢ (𝜃 → ¬ 𝜒)
2		nsyli.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	con3d 149	. 2 ⊢ (𝜑 → (¬ 𝜒 → ¬ 𝜓))
4	1, 3	syl5 34	1 ⊢ (𝜑 → (𝜃 → ¬ 𝜓))

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:  necon3ad  2950  tz7.7  5934  onssneli  6017  tz7.48-2  7741  tz7.49  7744  php  8351  nndomo  8361  elirrv  8708  setind  8825  zorn2lem3  9573  alephval2  9647  inar1  9850  dfon2lem6  32068  sltres  32191  finminlem  32688  onint1  32819  poimirlem4  33769  gneispace  39038