Theorem syl3an12 43006

Description: A double syllogism inference. (Contributed by SN, 15-Sep-2024.)

Hypotheses

Ref	Expression
syl3an12.1	⊢ (𝜑 → 𝜓)
syl3an12.2	⊢ (𝜒 → 𝜃)
syl3an12.s	⊢ ((𝜓 ∧ 𝜃 ∧ 𝜏) → 𝜂)

Assertion

Ref	Expression
syl3an12	⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜂)

Proof of Theorem syl3an12

Step	Hyp	Ref	Expression
1		syl3an12.1	. 2 ⊢ (𝜑 → 𝜓)
2		syl3an12.2	. 2 ⊢ (𝜒 → 𝜃)
3		id 23	. 2 ⊢ (𝜏 → 𝜏)
4		syl3an12.s	. 2 ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜏) → 𝜂)
5	1, 2, 3, 4	syl3an 1178	1 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜂)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103

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

This proof depends on definitions:  df-bi 210  df-an 401  df-3an 1105

This theorem is used by:  dvdsexpb  43124