Theorem cbvexfo

Description: Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997)

	Ref	Expression
Hypothesis	cbvfo.1	$$⊢F\left(x\right)=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	cbvexfo	$$⊢F:A\underset{onto}{⟶}B\to \left(\exists x\in A\phi \leftrightarrow \exists y\in B\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		cbvfo.1	$$⊢F\left(x\right)=y\to \left(\phi \leftrightarrow \psi \right)$$
2	1	notbid	$$⊢F\left(x\right)=y\to \left(\neg \phi \leftrightarrow \neg \psi \right)$$
3	2	cbvfo	$$⊢F:A\underset{onto}{⟶}B\to \left(\forall x\in A\neg \phi \leftrightarrow \forall y\in B\neg \psi \right)$$
4	3	notbid	$$⊢F:A\underset{onto}{⟶}B\to \left(\neg \forall x\in A\neg \phi \leftrightarrow \neg \forall y\in B\neg \psi \right)$$
5		dfrex2	$$⊢\exists x\in A\phi \leftrightarrow \neg \forall x\in A\neg \phi$$
6		dfrex2	$$⊢\exists y\in B\psi \leftrightarrow \neg \forall y\in B\neg \psi$$
7	4 5 6	3bitr4g	$$⊢F:A\underset{onto}{⟶}B\to \left(\exists x\in A\phi \leftrightarrow \exists y\in B\psi \right)$$