Theorem nfbiit

Description: Equivalence theorem for the non-freeness predicate. Closed form of nfbii . (Contributed by Giovanni Mascellani, 10-Apr-2018) Reduce axiom usage. (Revised by BJ, 6-May-2019)

	Ref	Expression
Assertion	nfbiit	$$⊢\forall x\left(\phi \leftrightarrow \psi \right)\to \left(Ⅎx\phi \leftrightarrow Ⅎx\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		exbi	$$⊢\forall x\left(\phi \leftrightarrow \psi \right)\to \left(\exists x\phi \leftrightarrow \exists x\psi \right)$$
2		albi	$$⊢\forall x\left(\phi \leftrightarrow \psi \right)\to \left(\forall x\phi \leftrightarrow \forall x\psi \right)$$
3	1 2	imbi12d	$$⊢\forall x\left(\phi \leftrightarrow \psi \right)\to \left(\left(\exists x\phi \to \forall x\phi \right)\leftrightarrow \left(\exists x\psi \to \forall x\psi \right)\right)$$
4		df-nf	$$⊢Ⅎx\phi \leftrightarrow \left(\exists x\phi \to \forall x\phi \right)$$
5		df-nf	$$⊢Ⅎx\psi \leftrightarrow \left(\exists x\psi \to \forall x\psi \right)$$
6	3 4 5	3bitr4g	$$⊢\forall x\left(\phi \leftrightarrow \psi \right)\to \left(Ⅎx\phi \leftrightarrow Ⅎx\psi \right)$$