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Metamath Proof Explorer

Theorem eel2131

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

Ref Expression
Hypotheses eel2131.1 ⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
eel2131.2 ⊢ ( ( 𝜑 ∧ 𝜃 ) → 𝜏 )
eel2131.3 ⊢ ( ( 𝜒 ∧ 𝜏 ) → 𝜂 )
Assertion eel2131 ( ( 𝜑 ∧ 𝜓 ∧ 𝜃 ) → 𝜂 )

Proof

Step Hyp Ref Expression
1 eel2131.1 ⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
2 eel2131.2 ⊢ ( ( 𝜑 ∧ 𝜃 ) → 𝜏 )
3 eel2131.3 ⊢ ( ( 𝜒 ∧ 𝜏 ) → 𝜂 )
4 1 2 3 syl2an ⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝜑 ∧ 𝜃 ) ) → 𝜂 )
5 4 3impdi ⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜃 ) → 𝜂 )