Theorem nfbi

Description: If x is not free in ph and ps , then it is not free in ( ph <-> ps ) . (Contributed by NM, 26-May-1993) (Revised by Mario Carneiro, 11-Aug-2016) (Proof shortened by Wolf Lammen, 2-Jan-2018)

	Ref	Expression
Hypotheses	nf.1	$$⊢Ⅎx\phi$$
	nf.2	$$⊢Ⅎx\psi$$
Assertion	nfbi	$$⊢Ⅎx\left(\phi \leftrightarrow \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		nf.1	$$⊢Ⅎx\phi$$
2		nf.2	$$⊢Ⅎx\psi$$
3	1	a1i	$$⊢\top \to Ⅎx\phi$$
4	2	a1i	$$⊢\top \to Ⅎx\psi$$
5	3 4	nfbid	$$⊢\top \to Ⅎx\left(\phi \leftrightarrow \psi \right)$$
6	5	mptru	$$⊢Ⅎx\left(\phi \leftrightarrow \psi \right)$$