Theorem nsyli 160

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 155	. 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  2947  tz7.7  6198  onssneli  6282  tz7.48-2  8107  tz7.49  8110  php  8751  nndomog  8758  elirrv  9133  setind  9249  zorn2lem3  9998  alephval2  10072  inar1  10275  dfon2lem6  33336  sltres  33506  finminlem  34145  onint1  34276  poimirlem4  35404  gneispace  41290