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Metamath Proof Explorer

Theorem syl2an3an

Description: syl3an with antecedents in standard conjunction form. (Contributed by Alan Sare, 31-Aug-2016)

Ref Expression
Hypotheses syl2an3an.1 ⊢ ( 𝜑 → 𝜓 )
syl2an3an.2 ⊢ ( 𝜑 → 𝜒 )
syl2an3an.3 ⊢ ( 𝜃 → 𝜏 )
syl2an3an.4 ⊢ ( ( 𝜓 ∧ 𝜒 ∧ 𝜏 ) → 𝜂 )
Assertion syl2an3an ( ( 𝜑 ∧ 𝜃 ) → 𝜂 )

Proof

Step Hyp Ref Expression
1 syl2an3an.1 ⊢ ( 𝜑 → 𝜓 )
2 syl2an3an.2 ⊢ ( 𝜑 → 𝜒 )
3 syl2an3an.3 ⊢ ( 𝜃 → 𝜏 )
4 syl2an3an.4 ⊢ ( ( 𝜓 ∧ 𝜒 ∧ 𝜏 ) → 𝜂 )
5 1 2 3 4 syl3an ⊢ ( ( 𝜑 ∧ 𝜑 ∧ 𝜃 ) → 𝜂 )
6 5 3anidm12 ⊢ ( ( 𝜑 ∧ 𝜃 ) → 𝜂 )