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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 → 𝐴 = 𝐵 )
	eqneltrd.2	⊢ ( 𝜑 → ¬ 𝐵 ∈ 𝐶 )
Assertion	eqneltrd	⊢ ( 𝜑 → ¬ 𝐴 ∈ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		eqneltrd.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		eqneltrd.2	⊢ ( 𝜑 → ¬ 𝐵 ∈ 𝐶 )
3	1	eleq1d	⊢ ( 𝜑 → ( 𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶 ) )
4	2 3	mtbird	⊢ ( 𝜑 → ¬ 𝐴 ∈ 𝐶 )