Theorem rexbi

Description: Distribute restricted quantification over a biconditional. (Contributed by Scott Fenton, 7-Aug-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		ralbi	$$⊢\forall x\in A\left(\neg \phi \leftrightarrow \neg \psi \right)\to \left(\forall x\in A\neg \phi \leftrightarrow \forall x\in A\neg \psi \right)$$
2	1	notbid	$$⊢\forall x\in A\left(\neg \phi \leftrightarrow \neg \psi \right)\to \left(\neg \forall x\in A\neg \phi \leftrightarrow \neg \forall x\in A\neg \psi \right)$$
3		notbi	$$⊢\left(\phi \leftrightarrow \psi \right)\leftrightarrow \left(\neg \phi \leftrightarrow \neg \psi \right)$$
4	3	ralbii	$$⊢\forall x\in A\left(\phi \leftrightarrow \psi \right)\leftrightarrow \forall x\in A\left(\neg \phi \leftrightarrow \neg \psi \right)$$
5		dfrex2	$$⊢\exists x\in A\phi \leftrightarrow \neg \forall x\in A\neg \phi$$
6		dfrex2	$$⊢\exists x\in A\psi \leftrightarrow \neg \forall x\in A\neg \psi$$
7	5 6	bibi12i	$$⊢\left(\exists x\in A\phi \leftrightarrow \exists x\in A\psi \right)\leftrightarrow \left(\neg \forall x\in A\neg \phi \leftrightarrow \neg \forall x\in A\neg \psi \right)$$
8	2 4 7	3imtr4i	$$⊢\forall x\in A\left(\phi \leftrightarrow \psi \right)\to \left(\exists x\in A\phi \leftrightarrow \exists x\in A\psi \right)$$