Theorem syl3anl 1284

Description: A triple syllogism inference. (Contributed by NM, 24-Dec-2006.)

Hypotheses

Ref	Expression
syl3anl.1	⊢ (𝜑 → 𝜓)
syl3anl.2	⊢ (𝜒 → 𝜃)
syl3anl.3	⊢ (𝜏 → 𝜂)
syl3anl.4	⊢ (((𝜓 ∧ 𝜃 ∧ 𝜂) ∧ 𝜁) → 𝜎)

Assertion

Ref	Expression
syl3anl	⊢ (((𝜑 ∧ 𝜒 ∧ 𝜏) ∧ 𝜁) → 𝜎)

Proof of Theorem syl3anl

Step	Hyp	Ref	Expression
1		syl3anl.1	. . 3 ⊢ (𝜑 → 𝜓)
2		syl3anl.2	. . 3 ⊢ (𝜒 → 𝜃)
3		syl3anl.3	. . 3 ⊢ (𝜏 → 𝜂)
4	1, 2, 3	3anim123i 1179	. 2 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → (𝜓 ∧ 𝜃 ∧ 𝜂))
5		syl3anl.4	. 2 ⊢ (((𝜓 ∧ 𝜃 ∧ 𝜂) ∧ 𝜁) → 𝜎)
6	4, 5	sylan 281	1 ⊢ (((𝜑 ∧ 𝜒 ∧ 𝜏) ∧ 𝜁) → 𝜎)

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116  df-3an 975

This theorem is referenced by: (None)