Theorem syl3anl 1442

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 1169	. 2 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → (𝜓 ∧ 𝜃 ∧ 𝜂))
5		syl3anl.4	. 2 ⊢ (((𝜓 ∧ 𝜃 ∧ 𝜂) ∧ 𝜁) → 𝜎)
6	4, 5	sylan 592	1 ⊢ (((𝜑 ∧ 𝜒 ∧ 𝜏) ∧ 𝜁) → 𝜎)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ 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 402  df-3an 1105

This theorem is used by:  rngridlmcl  21372  chlej1  31909  chlej2  31910  atcvatlem  32784  fzunt  44214  3f1oss2  47846