Theorem eleq1a

Description: A transitive-type law relating membership and equality. (Contributed by NM, 9-Apr-1994)

	Ref	Expression
Assertion	eleq1a	⊢ ( 𝐴 ∈ 𝐵 → ( 𝐶 = 𝐴 → 𝐶 ∈ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		eleq1	⊢ ( 𝐶 = 𝐴 → ( 𝐶 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵 ) )
2	1	biimprcd	⊢ ( 𝐴 ∈ 𝐵 → ( 𝐶 = 𝐴 → 𝐶 ∈ 𝐵 ) )