Theorem nfifd

Description: Deduction form of nfif . (Contributed by NM, 15-Feb-2013) (Revised by Mario Carneiro, 13-Oct-2016)

	Ref	Expression
Hypotheses	nfifd.2	$$⊢\phi \to Ⅎx\psi$$
	nfifd.3	$$⊢\phi \to \underset{\_}{Ⅎ}xA$$
	nfifd.4	$$⊢\phi \to \underset{\_}{Ⅎ}xB$$
Assertion	nfifd	$$⊢\phi \to \underset{\_}{Ⅎ}xif\left(\psi ,A,B\right)$$

Proof

Step	Hyp	Ref	Expression
1		nfifd.2	$$⊢\phi \to Ⅎx\psi$$
2		nfifd.3	$$⊢\phi \to \underset{\_}{Ⅎ}xA$$
3		nfifd.4	$$⊢\phi \to \underset{\_}{Ⅎ}xB$$
4		dfif2	$$⊢if\left(\psi ,A,B\right)=\left\{y|\left(\left(y\in B\to \psi \right)\to \left(y\in A\wedge \psi \right)\right)\right\}$$
5		nfv	$$⊢Ⅎy\phi$$
6	3	nfcrd	$$⊢\phi \to Ⅎxy\in B$$
7	6 1	nfimd	$$⊢\phi \to Ⅎx\left(y\in B\to \psi \right)$$
8	2	nfcrd	$$⊢\phi \to Ⅎxy\in A$$
9	8 1	nfand	$$⊢\phi \to Ⅎx\left(y\in A\wedge \psi \right)$$
10	7 9	nfimd	$$⊢\phi \to Ⅎx\left(\left(y\in B\to \psi \right)\to \left(y\in A\wedge \psi \right)\right)$$
11	5 10	nfabdw	$$⊢\phi \to \underset{\_}{Ⅎ}x\left\{y|\left(\left(y\in B\to \psi \right)\to \left(y\in A\wedge \psi \right)\right)\right\}$$
12	4 11	nfcxfrd	$$⊢\phi \to \underset{\_}{Ⅎ}xif\left(\psi ,A,B\right)$$