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Metamath Proof Explorer

Theorem eqneltrd

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)

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

Proof

Step	Hyp	Ref	Expression
1		eqneltrd.1	$⊢ φ \to A = B$
2		eqneltrd.2	$⊢ φ \to \neg B \in C$
3	1	eleq1d	$⊢ φ \to (A \in C \leftrightarrow B \in C)$
4	2 3	mtbird	$⊢ φ \to \neg A \in C$