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Metamath Proof Explorer

Theorem eel3132

Description: syl2an with antecedents in standard conjunction form. (Contributed by Alan Sare, 27-Aug-2016)

Ref Expression
Hypotheses eel3132.1 ⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
eel3132.2 ⊢ ( ( 𝜃 ∧ 𝜓 ) → 𝜏 )
eel3132.3 ⊢ ( ( 𝜒 ∧ 𝜏 ) → 𝜂 )
Assertion eel3132 ( ( 𝜑 ∧ 𝜃 ∧ 𝜓 ) → 𝜂 )

Proof

Step Hyp Ref Expression
1 eel3132.1 ⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
2 eel3132.2 ⊢ ( ( 𝜃 ∧ 𝜓 ) → 𝜏 )
3 eel3132.3 ⊢ ( ( 𝜒 ∧ 𝜏 ) → 𝜂 )
4 1 2 3 syl2an ⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝜃 ∧ 𝜓 ) ) → 𝜂 )
5 4 3impdir ⊢ ( ( 𝜑 ∧ 𝜃 ∧ 𝜓 ) → 𝜂 )