Theorem eqnetrrid

Description: A chained equality inference for inequality. (Contributed by NM, 6-Jun-2012) (Proof shortened by Wolf Lammen, 19-Nov-2019)

	Ref	Expression
Hypotheses	eqnetrrid.1	⊢ 𝐵 = 𝐴
	eqnetrrid.2	⊢ ( 𝜑 → 𝐵 ≠ 𝐶 )
Assertion	eqnetrrid	⊢ ( 𝜑 → 𝐴 ≠ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		eqnetrrid.1	⊢ 𝐵 = 𝐴
2		eqnetrrid.2	⊢ ( 𝜑 → 𝐵 ≠ 𝐶 )
3	1	a1i	⊢ ( 𝜑 → 𝐵 = 𝐴 )
4	3 2	eqnetrrd	⊢ ( 𝜑 → 𝐴 ≠ 𝐶 )