Theorem sbcex

Description: By our definition of proper substitution, it can only be true if the substituted expression is a set. (Contributed by Mario Carneiro, 13-Oct-2016)

	Ref	Expression
Assertion	sbcex	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to A\in \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		df-sbc	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow A\in \left\{x|\phi \right\}$$
2		elex	$$⊢A\in \left\{x|\phi \right\}\to A\in \mathrm{V}$$
3	1 2	sylbi	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to A\in \mathrm{V}$$