Theorem nsyld 159

Description: A negated syllogism deduction. (Contributed by NM, 9-Apr-2005.)

Hypotheses

Ref	Expression
nsyld.1	⊢ (𝜑 → (𝜓 → ¬ 𝜒))
nsyld.2	⊢ (𝜑 → (𝜏 → 𝜒))

Assertion

Ref	Expression
nsyld	⊢ (𝜑 → (𝜓 → ¬ 𝜏))

Proof of Theorem nsyld

Step	Hyp	Ref	Expression
1		nsyld.1	. 2 ⊢ (𝜑 → (𝜓 → ¬ 𝜒))
2		nsyld.2	. . 3 ⊢ (𝜑 → (𝜏 → 𝜒))
3	2	con3d 155	. 2 ⊢ (𝜑 → (¬ 𝜒 → ¬ 𝜏))
4	1, 3	syld 47	1 ⊢ (𝜑 → (𝜓 → ¬ 𝜏))

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:  pm2.65d  199  pltn2lp  17847  alexsubALTlem4  22947  eupth2eucrct  28300  ifeqeqx  30604  noinfbnd1lem1  33663  cvrat  37173  radcnvrat  41605