Theorem nfreud

Description: Deduction version of nfreu . Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 15-Feb-2013) (Revised by Mario Carneiro, 8-Oct-2016) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nfreud.1	$$⊢Ⅎy\phi$$
	nfreud.2	$$⊢\phi \to \underset{\_}{Ⅎ}xA$$
	nfreud.3	$$⊢\phi \to Ⅎx\psi$$
Assertion	nfreud	$$⊢\phi \to Ⅎx\exists !y\in A\psi$$

Proof

Step	Hyp	Ref	Expression
1		nfreud.1	$$⊢Ⅎy\phi$$
2		nfreud.2	$$⊢\phi \to \underset{\_}{Ⅎ}xA$$
3		nfreud.3	$$⊢\phi \to Ⅎx\psi$$
4		df-reu	$$⊢\exists !y\in A\psi \leftrightarrow \exists !y\left(y\in A\wedge \psi \right)$$
5		nfcvf	$$⊢\neg \forall xx=y\to \underset{\_}{Ⅎ}xy$$
6	5	adantl	$$⊢\left(\phi \wedge \neg \forall xx=y\right)\to \underset{\_}{Ⅎ}xy$$
7	2	adantr	$$⊢\left(\phi \wedge \neg \forall xx=y\right)\to \underset{\_}{Ⅎ}xA$$
8	6 7	nfeld	$$⊢\left(\phi \wedge \neg \forall xx=y\right)\to Ⅎxy\in A$$
9	3	adantr	$$⊢\left(\phi \wedge \neg \forall xx=y\right)\to Ⅎx\psi$$
10	8 9	nfand	$$⊢\left(\phi \wedge \neg \forall xx=y\right)\to Ⅎx\left(y\in A\wedge \psi \right)$$
11	1 10	nfeud2	$$⊢\phi \to Ⅎx\exists !y\left(y\in A\wedge \psi \right)$$
12	4 11	nfxfrd	$$⊢\phi \to Ⅎx\exists !y\in A\psi$$