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	$$⊢\phi \to A=B$$
	eqnbrtrd.2	$$⊢\phi \to \neg BRC$$
Assertion	eqnbrtrd	$$⊢\phi \to \neg ARC$$

Proof

Step	Hyp	Ref	Expression
1		eqnbrtrd.1	$$⊢\phi \to A=B$$
2		eqnbrtrd.2	$$⊢\phi \to \neg BRC$$
3	1	breq1d	$$⊢\phi \to \left(ARC\leftrightarrow BRC\right)$$
4	2 3	mtbird	$$⊢\phi \to \neg ARC$$