Theorem rexeqbidvv

Description: Version of rexeqbidv with additional disjoint variable conditions, not requiring ax-8 nor df-clel . (Contributed by Wolf Lammen, 25-Sep-2024)

	Ref	Expression
Hypotheses	raleqbidvv.1	$$⊢\phi \to A=B$$
	raleqbidvv.2	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
Assertion	rexeqbidvv	$$⊢\phi \to \left(\exists x\in A\psi \leftrightarrow \exists x\in B\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		raleqbidvv.1	$$⊢\phi \to A=B$$
2		raleqbidvv.2	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
3	2	notbid	$$⊢\phi \to \left(\neg \psi \leftrightarrow \neg \chi \right)$$
4	1 3	raleqbidvv	$$⊢\phi \to \left(\forall x\in A\neg \psi \leftrightarrow \forall x\in B\neg \chi \right)$$
5		ralnex	$$⊢\forall x\in A\neg \psi \leftrightarrow \neg \exists x\in A\psi$$
6		ralnex	$$⊢\forall x\in B\neg \chi \leftrightarrow \neg \exists x\in B\chi$$
7	4 5 6	3bitr3g	$$⊢\phi \to \left(\neg \exists x\in A\psi \leftrightarrow \neg \exists x\in B\chi \right)$$
8	7	con4bid	$$⊢\phi \to \left(\exists x\in A\psi \leftrightarrow \exists x\in B\chi \right)$$