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 )