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Metamath Proof Explorer

Theorem syl2anr

Description: A double syllogism inference. For an implication-only version, see syl2imc . (Contributed by NM, 17-Sep-2013)

Ref Expression
Hypotheses syl2an.1 ⊢ φ → ψ
syl2an.2 ⊢ τ → χ
syl2an.3 ⊢ ψ ∧ χ → θ
Assertion syl2anr ⊢ τ ∧ φ → θ

Proof

Step Hyp Ref Expression
1 syl2an.1 ⊢ φ → ψ
2 syl2an.2 ⊢ τ → χ
3 syl2an.3 ⊢ ψ ∧ χ → θ
4 1 2 3 syl2an ⊢ φ ∧ τ → θ
5 4 ancoms ⊢ τ ∧ φ → θ