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Metamath Proof Explorer

Theorem syl3anb

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

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

Proof

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