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 )