Theorem ralrexbid

Description: Formula-building rule for restricted existential quantifier, using a restricted universal quantifier to bind the quantified variable in the antecedent. (Contributed by AV, 21-Oct-2023) Reduce axiom usage. (Revised by SN, 13-Nov-2023)

	Ref	Expression
Hypothesis	ralrexbid.1	$$⊢\phi \to \left(\psi \leftrightarrow \theta \right)$$
Assertion	ralrexbid	$$⊢\forall x\in A\phi \to \left(\exists x\in A\psi \leftrightarrow \exists x\in A\theta \right)$$

Proof

Step	Hyp	Ref	Expression
1		ralrexbid.1	$$⊢\phi \to \left(\psi \leftrightarrow \theta \right)$$
2		df-ral	$$⊢\forall x\in A\phi \leftrightarrow \forall x\left(x\in A\to \phi \right)$$
3	1	imim2i	$$⊢\left(x\in A\to \phi \right)\to \left(x\in A\to \left(\psi \leftrightarrow \theta \right)\right)$$
4	3	pm5.32d	$$⊢\left(x\in A\to \phi \right)\to \left(\left(x\in A\wedge \psi \right)\leftrightarrow \left(x\in A\wedge \theta \right)\right)$$
5	4	alexbii	$$⊢\forall x\left(x\in A\to \phi \right)\to \left(\exists x\left(x\in A\wedge \psi \right)\leftrightarrow \exists x\left(x\in A\wedge \theta \right)\right)$$
6	2 5	sylbi	$$⊢\forall x\in A\phi \to \left(\exists x\left(x\in A\wedge \psi \right)\leftrightarrow \exists x\left(x\in A\wedge \theta \right)\right)$$
7		df-rex	$$⊢\exists x\in A\psi \leftrightarrow \exists x\left(x\in A\wedge \psi \right)$$
8		df-rex	$$⊢\exists x\in A\theta \leftrightarrow \exists x\left(x\in A\wedge \theta \right)$$
9	6 7 8	3bitr4g	$$⊢\forall x\in A\phi \to \left(\exists x\in A\psi \leftrightarrow \exists x\in A\theta \right)$$