Theorem rexrnmptw

Description: A restricted quantifier over an image set. Version of rexrnmpt with a disjoint variable condition, which does not require ax-13 . (Contributed by Mario Carneiro, 20-Aug-2015) (Revised by Gino Giotto, 26-Jan-2024)

	Ref	Expression
Hypotheses	rexrnmptw.1	$$⊢F=\left(x\in A⟼B\right)$$
	rexrnmptw.2	$$⊢y=B\to \left(\psi \leftrightarrow \chi \right)$$
Assertion	rexrnmptw	$$⊢\forall x\in AB\in V\to \left(\exists y\in \mathrm{ran}F\psi \leftrightarrow \exists x\in A\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		rexrnmptw.1	$$⊢F=\left(x\in A⟼B\right)$$
2		rexrnmptw.2	$$⊢y=B\to \left(\psi \leftrightarrow \chi \right)$$
3	2	notbid	$$⊢y=B\to \left(\neg \psi \leftrightarrow \neg \chi \right)$$
4	1 3	ralrnmptw	$$⊢\forall x\in AB\in V\to \left(\forall y\in \mathrm{ran}F\neg \psi \leftrightarrow \forall x\in A\neg \chi \right)$$
5	4	notbid	$$⊢\forall x\in AB\in V\to \left(\neg \forall y\in \mathrm{ran}F\neg \psi \leftrightarrow \neg \forall x\in A\neg \chi \right)$$
6		dfrex2	$$⊢\exists y\in \mathrm{ran}F\psi \leftrightarrow \neg \forall y\in \mathrm{ran}F\neg \psi$$
7		dfrex2	$$⊢\exists x\in A\chi \leftrightarrow \neg \forall x\in A\neg \chi$$
8	5 6 7	3bitr4g	$$⊢\forall x\in AB\in V\to \left(\exists y\in \mathrm{ran}F\psi \leftrightarrow \exists x\in A\chi \right)$$