Theorem syl10 79

Description: A nested syllogism inference. (Contributed by Alan Sare, 17-Jul-2011.)

Hypotheses

Ref	Expression
syl10.1	⊢ (𝜑 → (𝜓 → 𝜒))
syl10.2	⊢ (𝜑 → (𝜓 → (𝜃 → 𝜏)))
syl10.3	⊢ (𝜒 → (𝜏 → 𝜂))

Assertion

Ref	Expression
syl10	⊢ (𝜑 → (𝜓 → (𝜃 → 𝜂)))

Proof of Theorem syl10

Step	Hyp	Ref	Expression
1		syl10.2	. 2 ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜏)))
2		syl10.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3		syl10.3	. . 3 ⊢ (𝜒 → (𝜏 → 𝜂))
4	2, 3	syl6 35	. 2 ⊢ (𝜑 → (𝜓 → (𝜏 → 𝜂)))
5	1, 4	syldd 72	1 ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜂)))

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7

This theorem is referenced by:  tz7.49  8276  rspsbc2  42154  tratrb  42156