Theorem rspcef

Description: Restricted existential specialization, using implicit substitution. (Contributed by Glauco Siliprandi, 24-Dec-2020)

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

Proof

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