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Metamath Proof Explorer

Theorem eqneltrrd

Description: If a class is not an element of another class, an equal class is also not an element. Deduction form. (Contributed by David Moews, 1-May-2017) (Proof shortened by Wolf Lammen, 13-Nov-2019)

	Ref	Expression
Hypotheses	eqneltrrd.1	$⊢ φ \to A = B$
	eqneltrrd.2	$⊢ φ \to \neg A \in C$
Assertion	eqneltrrd	$⊢ φ \to \neg B \in C$

Proof

Step	Hyp	Ref	Expression
1		eqneltrrd.1	$⊢ φ \to A = B$
2		eqneltrrd.2	$⊢ φ \to \neg A \in C$
3	1	eqcomd	$⊢ φ \to B = A$
4	3 2	eqneltrd	$⊢ φ \to \neg B \in C$