Theorem rexrnmpt

Description: A restricted quantifier over an image set. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker rexrnmptw when possible. (Contributed by Mario Carneiro, 20-Aug-2015) (New usage is discouraged.)

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

Proof

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