Theorem rexbi

Description: Distribute restricted quantification over a biconditional. (Contributed by Scott Fenton, 7-Aug-2024) (Proof shortened by Wolf Lammen, 3-Nov-2024)

	Ref	Expression
Assertion	rexbi	$$⊢\forall x\in A\left(\phi \leftrightarrow \psi \right)\to \left(\exists x\in A\phi \leftrightarrow \exists x\in A\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		biimp	$$⊢\left(\phi \leftrightarrow \psi \right)\to \left(\phi \to \psi \right)$$
2	1	ralimi	$$⊢\forall x\in A\left(\phi \leftrightarrow \psi \right)\to \forall x\in A\left(\phi \to \psi \right)$$
3		rexim	$$⊢\forall x\in A\left(\phi \to \psi \right)\to \left(\exists x\in A\phi \to \exists x\in A\psi \right)$$
4	2 3	syl	$$⊢\forall x\in A\left(\phi \leftrightarrow \psi \right)\to \left(\exists x\in A\phi \to \exists x\in A\psi \right)$$
5		biimpr	$$⊢\left(\phi \leftrightarrow \psi \right)\to \left(\psi \to \phi \right)$$
6	5	ralimi	$$⊢\forall x\in A\left(\phi \leftrightarrow \psi \right)\to \forall x\in A\left(\psi \to \phi \right)$$
7		rexim	$$⊢\forall x\in A\left(\psi \to \phi \right)\to \left(\exists x\in A\psi \to \exists x\in A\phi \right)$$
8	6 7	syl	$$⊢\forall x\in A\left(\phi \leftrightarrow \psi \right)\to \left(\exists x\in A\psi \to \exists x\in A\phi \right)$$
9	4 8	impbid	$$⊢\forall x\in A\left(\phi \leftrightarrow \psi \right)\to \left(\exists x\in A\phi \leftrightarrow \exists x\in A\psi \right)$$