Theorem eqeltrd

Description: Substitution of equal classes into membership relation, deduction form. (Contributed by Raph Levien, 10-Dec-2002)

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

Proof

Step	Hyp	Ref	Expression
1		eqeltrd.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		eqeltrd.2	⊢ ( 𝜑 → 𝐵 ∈ 𝐶 )
3	1	eleq1d	⊢ ( 𝜑 → ( 𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶 ) )
4	2 3	mpbird	⊢ ( 𝜑 → 𝐴 ∈ 𝐶 )