Theorem eleq12

Description: Equality implies equivalence of membership. (Contributed by NM, 31-May-1999)

	Ref	Expression
Assertion	eleq12	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ( 𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		eleq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶 ) )
2		eleq2	⊢ ( 𝐶 = 𝐷 → ( 𝐵 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷 ) )
3	1 2	sylan9bb	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ( 𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷 ) )