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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