Theorem neleq12d

Description: Equality theorem for negated membership. (Contributed by FL, 10-Aug-2016) (Proof shortened by Wolf Lammen, 25-Nov-2019)

	Ref	Expression
Hypotheses	neleq12d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	neleq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
Assertion	neleq12d	⊢ ( 𝜑 → ( 𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		neleq12d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		neleq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
3	1 2	eleq12d	⊢ ( 𝜑 → ( 𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷 ) )
4	3	notbid	⊢ ( 𝜑 → ( ¬ 𝐴 ∈ 𝐶 ↔ ¬ 𝐵 ∈ 𝐷 ) )
5		df-nel	⊢ ( 𝐴 ∉ 𝐶 ↔ ¬ 𝐴 ∈ 𝐶 )
6		df-nel	⊢ ( 𝐵 ∉ 𝐷 ↔ ¬ 𝐵 ∈ 𝐷 )
7	4 5 6	3bitr4g	⊢ ( 𝜑 → ( 𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐷 ) )