Theorem eqnbrtrd

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

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

Proof

Step	Hyp	Ref	Expression
1		eqnbrtrd.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		eqnbrtrd.2	⊢ ( 𝜑 → ¬ 𝐵 𝑅 𝐶 )
3	1	breq1d	⊢ ( 𝜑 → ( 𝐴 𝑅 𝐶 ↔ 𝐵 𝑅 𝐶 ) )
4	2 3	mtbird	⊢ ( 𝜑 → ¬ 𝐴 𝑅 𝐶 )