Theorem nsyli 158

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 153	. 2 ⊢ (𝜑 → (¬ 𝜒 → ¬ 𝜓))
4	1, 3	syl5 35	1 ⊢ (𝜑 → (𝜃 → ¬ 𝜓))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is used by:  necon3ad  2971  tz7.7  6386  onssneli  6478  tz7.48-2  8425  tz7.49  8428  php  9187  elirrvOLDOLD  9557  setind  9712  zorn2lem3  10486  alephval2  10561  inar1  10764  ltsres  27835  setindregs  35551  dfon2lem6  36286  finminlem  36857  onint1  36988  poimirlem4  38303  ordnexbtwnsuc  44022  gneispace  44888