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Metamath Proof Explorer

Theorem syl3an132

Description: syl2an with antecedents in standard conjunction form. (Contributed by Alan Sare, 26-Aug-2016)

Ref Expression
Hypotheses syl3an132.1 ⊢ ( 𝜑 → 𝜓 )
syl3an132.2 ⊢ ( ( 𝜒 ∧ 𝜃 ) → 𝜏 )
syl3an132.3 ⊢ ( ( 𝜓 ∧ 𝜏 ) → 𝜂 )
Assertion syl3an132 ( ( 𝜑 ∧ 𝜒 ∧ 𝜃 ) → 𝜂 )

Proof

Step Hyp Ref Expression
1 syl3an132.1 ⊢ ( 𝜑 → 𝜓 )
2 syl3an132.2 ⊢ ( ( 𝜒 ∧ 𝜃 ) → 𝜏 )
3 syl3an132.3 ⊢ ( ( 𝜓 ∧ 𝜏 ) → 𝜂 )
4 1 2 3 syl2an ⊢ ( ( 𝜑 ∧ ( 𝜒 ∧ 𝜃 ) ) → 𝜂 )
5 4 3impb ⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜃 ) → 𝜂 )