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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 )