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Metamath Proof Explorer

Theorem 3imp3i2an

Description: An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017) (Proof shortened by Wolf Lammen, 13-Apr-2022)

Ref Expression
Hypotheses 3imp3i2an.1 ⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 )
3imp3i2an.2 ⊢ ( ( 𝜑 ∧ 𝜒 ) → 𝜏 )
3imp3i2an.3 ⊢ ( ( 𝜃 ∧ 𝜏 ) → 𝜂 )
Assertion 3imp3i2an ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜂 )

Proof

Step Hyp Ref Expression
1 3imp3i2an.1 ⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 )
2 3imp3i2an.2 ⊢ ( ( 𝜑 ∧ 𝜒 ) → 𝜏 )
3 3imp3i2an.3 ⊢ ( ( 𝜃 ∧ 𝜏 ) → 𝜂 )
4 2 3adant2 ⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜏 )
5 1 4 3 syl2anc ⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜂 )