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Metamath Proof Explorer

Theorem syl3anbrc

Description: Syllogism inference. (Contributed by Mario Carneiro, 11-May-2014)

Ref Expression
Hypotheses syl3anbrc.1 ⊢ ( 𝜑 → 𝜓 )
syl3anbrc.2 ⊢ ( 𝜑 → 𝜒 )
syl3anbrc.3 ⊢ ( 𝜑 → 𝜃 )
syl3anbrc.4 ⊢ ( 𝜏 ↔ ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) )
Assertion syl3anbrc ( 𝜑 → 𝜏 )

Proof

Step Hyp Ref Expression
1 syl3anbrc.1 ⊢ ( 𝜑 → 𝜓 )
2 syl3anbrc.2 ⊢ ( 𝜑 → 𝜒 )
3 syl3anbrc.3 ⊢ ( 𝜑 → 𝜃 )
4 syl3anbrc.4 ⊢ ( 𝜏 ↔ ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) )
5 1 2 3 3jca ⊢ ( 𝜑 → ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) )
6 5 4 sylibr ⊢ ( 𝜑 → 𝜏 )