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

Proof

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