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) (Proof shortened by Wolf Lammen, 4-Nov-2024)

	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	1	ralimi	$$⊢\forall x\in A\phi \to \forall x\in A\left(\psi \leftrightarrow \theta \right)$$
3		rexbi	$$⊢\forall x\in A\left(\psi \leftrightarrow \theta \right)\to \left(\exists x\in A\psi \leftrightarrow \exists x\in A\theta \right)$$
4	2 3	syl	$$⊢\forall x\in A\phi \to \left(\exists x\in A\psi \leftrightarrow \exists x\in A\theta \right)$$