Theorem eleq12i

Description: Inference from equality to equivalence of membership. (Contributed by NM, 31-May-1994)

	Ref	Expression
Hypotheses	eleq1i.1	⊢ 𝐴 = 𝐵
	eleq12i.2	⊢ 𝐶 = 𝐷
Assertion	eleq12i	⊢ ( 𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		eleq1i.1	⊢ 𝐴 = 𝐵
2		eleq12i.2	⊢ 𝐶 = 𝐷
3	2	eleq2i	⊢ ( 𝐴 ∈ 𝐶 ↔ 𝐴 ∈ 𝐷 )
4	1	eleq1i	⊢ ( 𝐴 ∈ 𝐷 ↔ 𝐵 ∈ 𝐷 )
5	3 4	bitri	⊢ ( 𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷 )