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Metamath Proof Explorer

Theorem syl22anbrc

Description: Syllogism inference. (Contributed by Thierry Arnoux, 19-Oct-2025)

Ref Expression
Hypotheses syl22anbrc.1 ⊢ ( 𝜑 → 𝜓 )
syl22anbrc.2 ⊢ ( 𝜑 → 𝜒 )
syl22anbrc.3 ⊢ ( 𝜑 → 𝜃 )
syl22anbrc.4 ⊢ ( 𝜑 → 𝜏 )
syl22anbrc.5 ⊢ ( 𝜂 ↔ ( ( 𝜓 ∧ 𝜒 ) ∧ ( 𝜃 ∧ 𝜏 ) ) )
Assertion syl22anbrc ( 𝜑 → 𝜂 )

Proof

Step Hyp Ref Expression
1 syl22anbrc.1 ⊢ ( 𝜑 → 𝜓 )
2 syl22anbrc.2 ⊢ ( 𝜑 → 𝜒 )
3 syl22anbrc.3 ⊢ ( 𝜑 → 𝜃 )
4 syl22anbrc.4 ⊢ ( 𝜑 → 𝜏 )
5 syl22anbrc.5 ⊢ ( 𝜂 ↔ ( ( 𝜓 ∧ 𝜒 ) ∧ ( 𝜃 ∧ 𝜏 ) ) )
6 3 4 jca ⊢ ( 𝜑 → ( 𝜃 ∧ 𝜏 ) )
7 1 2 6 5 syl21anbrc ⊢ ( 𝜑 → 𝜂 )