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Metamath Proof Explorer

Theorem syl2anc2

Description: Double syllogism inference combined with contraction. (Contributed by BTernaryTau, 29-Sep-2023)

Ref Expression
Hypotheses syl2anc2.1 ⊢ φ → ψ
syl2anc2.2 ⊢ ψ → χ
syl2anc2.3 ⊢ ψ ∧ χ → θ
Assertion syl2anc2 ⊢ φ → θ

Proof

Step Hyp Ref Expression
1 syl2anc2.1 ⊢ φ → ψ
2 syl2anc2.2 ⊢ ψ → χ
3 syl2anc2.3 ⊢ ψ ∧ χ → θ
4 1 2 syl ⊢ φ → χ
5 1 4 3 syl2anc ⊢ φ → θ