Theorem rspceaimv

Description: Restricted existential specialization of a universally quantified implication. (Contributed by BJ, 24-Aug-2022)

	Ref	Expression
Hypothesis	rspceaimv.1	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	rspceaimv	$$⊢\left(A\in B\wedge \forall y\in C\left(\psi \to \chi \right)\right)\to \exists x\in B\forall y\in C\left(\phi \to \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		rspceaimv.1	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
2	1	imbi1d	$$⊢x=A\to \left(\left(\phi \to \chi \right)\leftrightarrow \left(\psi \to \chi \right)\right)$$
3	2	ralbidv	$$⊢x=A\to \left(\forall y\in C\left(\phi \to \chi \right)\leftrightarrow \forall y\in C\left(\psi \to \chi \right)\right)$$
4	3	rspcev	$$⊢\left(A\in B\wedge \forall y\in C\left(\psi \to \chi \right)\right)\to \exists x\in B\forall y\in C\left(\phi \to \chi \right)$$