Theorem syl3anbr 1180

Description: A triple syllogism inference. (Contributed by NM, 29-Dec-2011.)

Hypotheses

Ref	Expression
syl3anbr.1	⊢ (𝜓 ↔ 𝜑)
syl3anbr.2	⊢ (𝜃 ↔ 𝜒)
syl3anbr.3	⊢ (𝜂 ↔ 𝜏)
syl3anbr.4	⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) → 𝜁)

Assertion

Ref	Expression
syl3anbr	⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜁)

Proof of Theorem syl3anbr

Step	Hyp	Ref	Expression
1		syl3anbr.1	. . 3 ⊢ (𝜓 ↔ 𝜑)
2	1	bicomi 227	. 2 ⊢ (𝜑 ↔ 𝜓)
3		syl3anbr.2	. . 3 ⊢ (𝜃 ↔ 𝜒)
4	3	bicomi 227	. 2 ⊢ (𝜒 ↔ 𝜃)
5		syl3anbr.3	. . 3 ⊢ (𝜂 ↔ 𝜏)
6	5	bicomi 227	. 2 ⊢ (𝜏 ↔ 𝜂)
7		syl3anbr.4	. 2 ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) → 𝜁)
8	2, 4, 6, 7	syl3anb 1179	1 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜁)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ 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:  colinearxfr  36762  paddval  40775