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Metamath Proof Explorer

Theorem sylancbr

Description: A syllogism inference combined with contraction. (Contributed by NM, 3-Sep-2004)

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

Proof

Step Hyp Ref Expression
1 sylancbr.1 ⊢ ψ ↔ φ
2 sylancbr.2 ⊢ χ ↔ φ
3 sylancbr.3 ⊢ ψ ∧ χ → θ
4 1 2 3 syl2anbr ⊢ φ ∧ φ → θ
5 4 anidms ⊢ φ → θ