Theorem syl6mpi 68

Description: A syllogism inference. (Contributed by Alan Sare, 8-Jul-2011.) (Proof shortened by Wolf Lammen, 13-Sep-2012.)

Hypotheses

Ref	Expression
syl6mpi.1	⊢ (𝜑 → (𝜓 → 𝜒))
syl6mpi.2	⊢ 𝜃
syl6mpi.3	⊢ (𝜒 → (𝜃 → 𝜏))

Assertion

Ref	Expression
syl6mpi	⊢ (𝜑 → (𝜓 → 𝜏))

Proof of Theorem syl6mpi

Step	Hyp	Ref	Expression
1		syl6mpi.1	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
2		syl6mpi.2	. . 3 ⊢ 𝜃
3		syl6mpi.3	. . 3 ⊢ (𝜒 → (𝜃 → 𝜏))
4	2, 3	mpi 21	. 2 ⊢ (𝜒 → 𝜏)
5	1, 4	syl6 36	1 ⊢ (𝜑 → (𝜓 → 𝜏))

This proof depends on syntax axioms:   → wi 4

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

This theorem is used by:  axc15  2454  bndrank  9809  ac10ct  10023  1re  11212  fvf1tp  13827  uspgrn2crct  30166  tratrb  45273  ee20an  45465