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  2956  tz7.7  6292  onssneli  6376  tz7.48-2  8273  tz7.49  8276  php  8993  phpOLD  9005  nndomogOLD  9009  elirrv  9355  setind  9492  zorn2lem3  10254  alephval2  10328  inar1  10531  dfon2lem6  33764  sltres  33865  finminlem  34507  onint1  34638  poimirlem4  35781  gneispace  41744