Theorem reubidv

Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 17-Oct-1996)

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

Proof

Step	Hyp	Ref	Expression
1		reubidv.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
2	1	adantr	$$⊢\left(\phi \wedge x\in A\right)\to \left(\psi \leftrightarrow \chi \right)$$
3	2	reubidva	$$⊢\phi \to \left(\exists !x\in A\psi \leftrightarrow \exists !x\in A\chi \right)$$