Theorem sylan9req

Description: An equality transitivity deduction. (Contributed by NM, 23-Jun-2007)

	Ref	Expression
Hypotheses	sylan9req.1	⊢ ( 𝜑 → 𝐵 = 𝐴 )
	sylan9req.2	⊢ ( 𝜓 → 𝐵 = 𝐶 )
Assertion	sylan9req	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐴 = 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		sylan9req.1	⊢ ( 𝜑 → 𝐵 = 𝐴 )
2		sylan9req.2	⊢ ( 𝜓 → 𝐵 = 𝐶 )
3	1	eqcomd	⊢ ( 𝜑 → 𝐴 = 𝐵 )
4	3 2	sylan9eq	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐴 = 𝐶 )