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Metamath Proof Explorer

Theorem syl3anbr

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

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

Proof

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