Theorem sbcel21v

Description: Class substitution into a membership relation. One direction of sbcel2gv that holds for proper classes. (Contributed by NM, 17-Aug-2018)

	Ref	Expression
Assertion	sbcel21v	$$⊢\underset{\dot{}}{[}B/x\underset{\dot{}}{]}A\in x\to A\in B$$

Proof

Step	Hyp	Ref	Expression
1		sbcex	$$⊢\underset{\dot{}}{[}B/x\underset{\dot{}}{]}A\in x\to B\in \mathrm{V}$$
2		sbcel2gv	$$⊢B\in \mathrm{V}\to \left(\underset{\dot{}}{[}B/x\underset{\dot{}}{]}A\in x\leftrightarrow A\in B\right)$$
3	2	biimpd	$$⊢B\in \mathrm{V}\to \left(\underset{\dot{}}{[}B/x\underset{\dot{}}{]}A\in x\to A\in B\right)$$
4	1 3	mpcom	$$⊢\underset{\dot{}}{[}B/x\underset{\dot{}}{]}A\in x\to A\in B$$