Theorem nfif

Description: Bound-variable hypothesis builder for a conditional operator. (Contributed by NM, 16-Feb-2005) (Proof shortened by Andrew Salmon, 26-Jun-2011)

	Ref	Expression
Hypotheses	nfif.1	$$⊢Ⅎx\phi$$
	nfif.2	$$⊢\underset{\_}{Ⅎ}xA$$
	nfif.3	$$⊢\underset{\_}{Ⅎ}xB$$
Assertion	nfif	$$⊢\underset{\_}{Ⅎ}xif\left(\phi ,A,B\right)$$

Proof

Step	Hyp	Ref	Expression
1		nfif.1	$$⊢Ⅎx\phi$$
2		nfif.2	$$⊢\underset{\_}{Ⅎ}xA$$
3		nfif.3	$$⊢\underset{\_}{Ⅎ}xB$$
4	1	a1i	$$⊢\top \to Ⅎx\phi$$
5	2	a1i	$$⊢\top \to \underset{\_}{Ⅎ}xA$$
6	3	a1i	$$⊢\top \to \underset{\_}{Ⅎ}xB$$
7	4 5 6	nfifd	$$⊢\top \to \underset{\_}{Ⅎ}xif\left(\phi ,A,B\right)$$
8	7	mptru	$$⊢\underset{\_}{Ⅎ}xif\left(\phi ,A,B\right)$$