Theorem syl3anb 1178

Description: A triple syllogism inference. (Contributed by NM, 15-Oct-2005.)

Hypotheses

Ref	Expression
syl3anb.1	⊢ (𝜑 ↔ 𝜓)
syl3anb.2	⊢ (𝜒 ↔ 𝜃)
syl3anb.3	⊢ (𝜏 ↔ 𝜂)
syl3anb.4	⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) → 𝜁)

Assertion

Ref	Expression
syl3anb	⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜁)

Proof of Theorem syl3anb

Step	Hyp	Ref	Expression
1		syl3anb.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
2		syl3anb.2	. . 3 ⊢ (𝜒 ↔ 𝜃)
3		syl3anb.3	. . 3 ⊢ (𝜏 ↔ 𝜂)
4	1, 2, 3	3anbi123i 1093	. 2 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) ↔ (𝜓 ∧ 𝜃 ∧ 𝜂))
5		syl3anb.4	. 2 ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) → 𝜁)
6	4, 5	sylbi 114	1 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜁)

Syntax hints:   → wi 4   ↔ wb 98   ∧ w3a 885

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110  df-3an 887

This theorem is referenced by:  syl3anbr  1179  poxp  5853