Theorem sylan9eqr

Description: An equality transitivity deduction. (Contributed by NM, 8-May-1994)

	Ref	Expression
Hypotheses	sylan9eqr.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	sylan9eqr.2	⊢ ( 𝜓 → 𝐵 = 𝐶 )
Assertion	sylan9eqr	⊢ ( ( 𝜓 ∧ 𝜑 ) → 𝐴 = 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		sylan9eqr.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		sylan9eqr.2	⊢ ( 𝜓 → 𝐵 = 𝐶 )
3	1 2	sylan9eq	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐴 = 𝐶 )
4	3	ancoms	⊢ ( ( 𝜓 ∧ 𝜑 ) → 𝐴 = 𝐶 )