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	$$⊢\phi \to \neg ARB$$
	brneqtrd.2	$$⊢\phi \to B=C$$
Assertion	brneqtrd	$$⊢\phi \to \neg ARC$$

Proof

Step	Hyp	Ref	Expression
1		brneqtrd.1	$$⊢\phi \to \neg ARB$$
2		brneqtrd.2	$$⊢\phi \to B=C$$
3	2	breq2d	$$⊢\phi \to \left(ARB\leftrightarrow ARC\right)$$
4	1 3	mtbid	$$⊢\phi \to \neg ARC$$