Theorem nsyli 157

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 152	. 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  6337  onssneli  6428  tz7.48-2  8372  tz7.49  8375  php  9132  elirrvOLDOLD  9505  setind  9660  zorn2lem3  10412  alephval2  10487  inar1  10690  ltsres  27645  setindregs  35320  dfon2lem6  36023  finminlem  36555  onint1  36686  poimirlem4  38000  ordnexbtwnsuc  43721  gneispace  44587