Theorem reubidva

Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 13-Nov-2004) Reduce axiom usage. (Revised by Wolf Lammen, 14-Jan-2023)

	Ref	Expression
Hypothesis	reubidva.1	$$⊢\left(\phi \wedge x\in A\right)\to \left(\psi \leftrightarrow \chi \right)$$
Assertion	reubidva	$$⊢\phi \to \left(\exists !x\in A\psi \leftrightarrow \exists !x\in A\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		reubidva.1	$$⊢\left(\phi \wedge x\in A\right)\to \left(\psi \leftrightarrow \chi \right)$$
2	1	pm5.32da	$$⊢\phi \to \left(\left(x\in A\wedge \psi \right)\leftrightarrow \left(x\in A\wedge \chi \right)\right)$$
3	2	eubidv	$$⊢\phi \to \left(\exists !x\left(x\in A\wedge \psi \right)\leftrightarrow \exists !x\left(x\in A\wedge \chi \right)\right)$$
4		df-reu	$$⊢\exists !x\in A\psi \leftrightarrow \exists !x\left(x\in A\wedge \psi \right)$$
5		df-reu	$$⊢\exists !x\in A\chi \leftrightarrow \exists !x\left(x\in A\wedge \chi \right)$$
6	3 4 5	3bitr4g	$$⊢\phi \to \left(\exists !x\in A\psi \leftrightarrow \exists !x\in A\chi \right)$$