Theorem eldifsnd

Description: Membership in a set with an element removed : deduction version. (Contributed by Thierry Arnoux, 4-May-2025)

	Ref	Expression
Hypotheses	eldifsnd.1	$$⊢\phi \to A\in B$$
	eldifsnd.2	$$⊢\phi \to A\ne C$$
Assertion	eldifsnd	$$⊢\phi \to A\in \left(B\setminus \left\{C\right\}\right)$$

Proof

Step	Hyp	Ref	Expression
1		eldifsnd.1	$$⊢\phi \to A\in B$$
2		eldifsnd.2	$$⊢\phi \to A\ne C$$
3		eldifsn	$$⊢A\in \left(B\setminus \left\{C\right\}\right)\leftrightarrow \left(A\in B\wedge A\ne C\right)$$
4	1 2 3	sylanbrc	$$⊢\phi \to A\in \left(B\setminus \left\{C\right\}\right)$$