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  2943  tz7.7  6341  onssneli  6432  tz7.48-2  8371  tz7.49  8374  php  9129  elirrvOLD  9501  setind  9654  zorn2lem3  10406  alephval2  10481  inar1  10684  sltres  27628  setindregs  35235  dfon2lem6  35929  finminlem  36461  onint1  36592  poimirlem4  37764  ordnexbtwnsuc  43451  gneispace  44317