Theorem nfiotad

Description: Deduction version of nfiota . Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker nfiotadw when possible. (Contributed by NM, 18-Feb-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nfiotad.1	$$⊢Ⅎy\phi$$
	nfiotad.2	$$⊢\phi \to Ⅎx\psi$$
Assertion	nfiotad	$$⊢\phi \to \underset{\_}{Ⅎ}x\left(\iota y|\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		nfiotad.1	$$⊢Ⅎy\phi$$
2		nfiotad.2	$$⊢\phi \to Ⅎx\psi$$
3		dfiota2	$$⊢\left(\iota y|\psi \right)=\bigcup \left\{z|\forall y\left(\psi \leftrightarrow y=z\right)\right\}$$
4		nfv	$$⊢Ⅎz\phi$$
5	2	adantr	$$⊢\left(\phi \wedge \neg \forall xx=y\right)\to Ⅎx\psi$$
6		nfeqf1	$$⊢\neg \forall xx=y\to Ⅎxy=z$$
7	6	adantl	$$⊢\left(\phi \wedge \neg \forall xx=y\right)\to Ⅎxy=z$$
8	5 7	nfbid	$$⊢\left(\phi \wedge \neg \forall xx=y\right)\to Ⅎx\left(\psi \leftrightarrow y=z\right)$$
9	1 8	nfald2	$$⊢\phi \to Ⅎx\forall y\left(\psi \leftrightarrow y=z\right)$$
10	4 9	nfabd	$$⊢\phi \to \underset{\_}{Ⅎ}x\left\{z|\forall y\left(\psi \leftrightarrow y=z\right)\right\}$$
11	10	nfunid	$$⊢\phi \to \underset{\_}{Ⅎ}x\bigcup \left\{z|\forall y\left(\psi \leftrightarrow y=z\right)\right\}$$
12	3 11	nfcxfrd	$$⊢\phi \to \underset{\_}{Ⅎ}x\left(\iota y|\psi \right)$$