Theorem sylan9eq

Description: An equality transitivity deduction. (Contributed by NM, 8-May-1994) (Proof shortened by Andrew Salmon, 25-May-2011)

	Ref	Expression
Hypotheses	sylan9eq.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	sylan9eq.2	⊢ ( 𝜓 → 𝐵 = 𝐶 )
Assertion	sylan9eq	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐴 = 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		sylan9eq.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		sylan9eq.2	⊢ ( 𝜓 → 𝐵 = 𝐶 )
3		eqtr	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐵 = 𝐶 ) → 𝐴 = 𝐶 )
4	1 2 3	syl2an	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐴 = 𝐶 )