Theorem sbcrex

Description: Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005) (Revised by NM, 18-Aug-2018)

	Ref	Expression
Assertion	sbcrex	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\exists y\in B\phi \leftrightarrow \exists y\in B\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi$$

Proof

Step	Hyp	Ref	Expression
1		nfcv	$$⊢\underset{\_}{Ⅎ}yA$$
2		sbcrext	$$⊢\underset{\_}{Ⅎ}yA\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\exists y\in B\phi \leftrightarrow \exists y\in B\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)$$
3	1 2	ax-mp	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\exists y\in B\phi \leftrightarrow \exists y\in B\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi$$