Theorem brneqtrd

Description: Substitution of equal classes into the negation of a binary relation. (Contributed by Glauco Siliprandi, 3-Jan-2021)

	Ref	Expression
Hypotheses	brneqtrd.1	⊢ ( 𝜑 → ¬ 𝐴 𝑅 𝐵 )
	brneqtrd.2	⊢ ( 𝜑 → 𝐵 = 𝐶 )
Assertion	brneqtrd	⊢ ( 𝜑 → ¬ 𝐴 𝑅 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		brneqtrd.1	⊢ ( 𝜑 → ¬ 𝐴 𝑅 𝐵 )
2		brneqtrd.2	⊢ ( 𝜑 → 𝐵 = 𝐶 )
3	2	breq2d	⊢ ( 𝜑 → ( 𝐴 𝑅 𝐵 ↔ 𝐴 𝑅 𝐶 ) )
4	1 3	mtbid	⊢ ( 𝜑 → ¬ 𝐴 𝑅 𝐶 )