Theorem rspced

Description: Restricted existential specialization, using implicit substitution. (Contributed by Glauco Siliprandi, 15-Feb-2025)

	Ref	Expression
Hypotheses	rspced.1	$$⊢Ⅎx\chi$$
	rspced.2	$$⊢\underset{\_}{Ⅎ}xA$$
	rspced.3	$$⊢\underset{\_}{Ⅎ}xB$$
	rspced.4	$$⊢\phi \to A\in B$$
	rspced.5	$$⊢\phi \to \chi$$
	rspced.6	$$⊢x=A\to \left(\psi \leftrightarrow \chi \right)$$
Assertion	rspced	$$⊢\phi \to \exists x\in B\psi$$

Proof

Step	Hyp	Ref	Expression
1		rspced.1	$$⊢Ⅎx\chi$$
2		rspced.2	$$⊢\underset{\_}{Ⅎ}xA$$
3		rspced.3	$$⊢\underset{\_}{Ⅎ}xB$$
4		rspced.4	$$⊢\phi \to A\in B$$
5		rspced.5	$$⊢\phi \to \chi$$
6		rspced.6	$$⊢x=A\to \left(\psi \leftrightarrow \chi \right)$$
7	1 2 3 6	rspcef	$$⊢\left(A\in B\wedge \chi \right)\to \exists x\in B\psi$$
8	4 5 7	syl2anc	$$⊢\phi \to \exists x\in B\psi$$