Theorem eleq12d

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

	Ref	Expression
Hypotheses	eleq12d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	eleq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
Assertion	eleq12d	⊢ ( 𝜑 → ( 𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		eleq12d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		eleq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
3	2	eleq2d	⊢ ( 𝜑 → ( 𝐴 ∈ 𝐶 ↔ 𝐴 ∈ 𝐷 ) )
4	1	eleq1d	⊢ ( 𝜑 → ( 𝐴 ∈ 𝐷 ↔ 𝐵 ∈ 𝐷 ) )
5	3 4	bitrd	⊢ ( 𝜑 → ( 𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷 ) )