Theorem reubiia

Description: Formula-building rule for restricted existential quantifier (inference form). (Contributed by NM, 14-Nov-2004)

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

Proof

Step	Hyp	Ref	Expression
1		reubiia.1	$$⊢x\in A\to \left(\phi \leftrightarrow \psi \right)$$
2	1	pm5.32i	$$⊢\left(x\in A\wedge \phi \right)\leftrightarrow \left(x\in A\wedge \psi \right)$$
3	2	eubii	$$⊢\exists !x\left(x\in A\wedge \phi \right)\leftrightarrow \exists !x\left(x\in A\wedge \psi \right)$$
4		df-reu	$$⊢\exists !x\in A\phi \leftrightarrow \exists !x\left(x\in A\wedge \phi \right)$$
5		df-reu	$$⊢\exists !x\in A\psi \leftrightarrow \exists !x\left(x\in A\wedge \psi \right)$$
6	3 4 5	3bitr4i	$$⊢\exists !x\in A\phi \leftrightarrow \exists !x\in A\psi$$