Theorem eqeltrd

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

	Ref	Expression
Hypotheses	eqeltrd.1	$$⊢\phi \to A=B$$
	eqeltrd.2	$$⊢\phi \to B\in C$$
Assertion	eqeltrd	$$⊢\phi \to A\in C$$

Proof

Step	Hyp	Ref	Expression
1		eqeltrd.1	$$⊢\phi \to A=B$$
2		eqeltrd.2	$$⊢\phi \to B\in C$$
3	1	eleq1d	$$⊢\phi \to \left(A\in C\leftrightarrow B\in C\right)$$
4	2 3	mpbird	$$⊢\phi \to A\in C$$