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Metamath Proof Explorer

Theorem sylancb

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

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

Proof

Step Hyp Ref Expression
1 sylancb.1 ⊢ φ ↔ ψ
2 sylancb.2 ⊢ φ ↔ χ
3 sylancb.3 ⊢ ψ ∧ χ → θ
4 1 2 3 syl2anb ⊢ φ ∧ φ → θ
5 4 anidms ⊢ φ → θ