This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem 3syld

Description: Triple syllogism deduction. Deduction associated with 3syld . (Contributed by Jeff Hankins, 4-Aug-2009)

Ref Expression
Hypotheses 3syld.1 ⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
3syld.2 ⊢ ( 𝜑 → ( 𝜒 → 𝜃 ) )
3syld.3 ⊢ ( 𝜑 → ( 𝜃 → 𝜏 ) )
Assertion 3syld ( 𝜑 → ( 𝜓 → 𝜏 ) )

Proof

Step Hyp Ref Expression
1 3syld.1 ⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2 3syld.2 ⊢ ( 𝜑 → ( 𝜒 → 𝜃 ) )
3 3syld.3 ⊢ ( 𝜑 → ( 𝜃 → 𝜏 ) )
4 1 2 syld ⊢ ( 𝜑 → ( 𝜓 → 𝜃 ) )
5 4 3 syld ⊢ ( 𝜑 → ( 𝜓 → 𝜏 ) )