Theorem rspcebdv

Description: Restricted existential specialization, using implicit substitution in both directions. (Contributed by AV, 8-Jan-2022)

	Ref	Expression
Hypotheses	rspcdv.1	$$⊢\phi \to A\in B$$
	rspcdv.2	$$⊢\left(\phi \wedge x=A\right)\to \left(\psi \leftrightarrow \chi \right)$$
	rspcebdv.1	$$⊢\left(\phi \wedge \psi \right)\to x=A$$
Assertion	rspcebdv	$$⊢\phi \to \left(\exists x\in B\psi \leftrightarrow \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		rspcdv.1	$$⊢\phi \to A\in B$$
2		rspcdv.2	$$⊢\left(\phi \wedge x=A\right)\to \left(\psi \leftrightarrow \chi \right)$$
3		rspcebdv.1	$$⊢\left(\phi \wedge \psi \right)\to x=A$$
4	3 2	syldan	$$⊢\left(\phi \wedge \psi \right)\to \left(\psi \leftrightarrow \chi \right)$$
5	4	biimpd	$$⊢\left(\phi \wedge \psi \right)\to \left(\psi \to \chi \right)$$
6	5	expcom	$$⊢\psi \to \left(\phi \to \left(\psi \to \chi \right)\right)$$
7	6	pm2.43b	$$⊢\phi \to \left(\psi \to \chi \right)$$
8	7	rexlimdvw	$$⊢\phi \to \left(\exists x\in B\psi \to \chi \right)$$
9	1 2	rspcedv	$$⊢\phi \to \left(\chi \to \exists x\in B\psi \right)$$
10	8 9	impbid	$$⊢\phi \to \left(\exists x\in B\psi \leftrightarrow \chi \right)$$