Theorem rexima

Description: Existential quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015) Reduce DV conditions. (Revised by Matthew House, 14-Aug-2025)

	Ref	Expression
Hypothesis	ralima.x	$$⊢x=F\left(y\right)\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	rexima	$$⊢\left(FFnA\wedge B\subseteq A\right)\to \left(\exists x\in F\left[B\right]\phi \leftrightarrow \exists y\in B\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		ralima.x	$$⊢x=F\left(y\right)\to \left(\phi \leftrightarrow \psi \right)$$
2	1	notbid	$$⊢x=F\left(y\right)\to \left(\neg \phi \leftrightarrow \neg \psi \right)$$
3	2	ralima	$$⊢\left(FFnA\wedge B\subseteq A\right)\to \left(\forall x\in F\left[B\right]\neg \phi \leftrightarrow \forall y\in B\neg \psi \right)$$
4	3	notbid	$$⊢\left(FFnA\wedge B\subseteq A\right)\to \left(\neg \forall x\in F\left[B\right]\neg \phi \leftrightarrow \neg \forall y\in B\neg \psi \right)$$
5		dfrex2	$$⊢\exists x\in F\left[B\right]\phi \leftrightarrow \neg \forall x\in F\left[B\right]\neg \phi$$
6		dfrex2	$$⊢\exists y\in B\psi \leftrightarrow \neg \forall y\in B\neg \psi$$
7	4 5 6	3bitr4g	$$⊢\left(FFnA\wedge B\subseteq A\right)\to \left(\exists x\in F\left[B\right]\phi \leftrightarrow \exists y\in B\psi \right)$$