Theorem nfbidv

Description: An equality theorem for nonfreeness. See nfbidf for a version without disjoint variable condition but requiring more axioms. (Contributed by Mario Carneiro, 4-Oct-2016) Remove dependency on ax-6 , ax-7 , ax-12 by adapting proof of nfbidf . (Revised by BJ, 25-Sep-2022)

	Ref	Expression
Hypothesis	albidv.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
Assertion	nfbidv	$$⊢\phi \to \left(Ⅎx\psi \leftrightarrow Ⅎx\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		albidv.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
2	1	exbidv	$$⊢\phi \to \left(\exists x\psi \leftrightarrow \exists x\chi \right)$$
3	1	albidv	$$⊢\phi \to \left(\forall x\psi \leftrightarrow \forall x\chi \right)$$
4	2 3	imbi12d	$$⊢\phi \to \left(\left(\exists x\psi \to \forall x\psi \right)\leftrightarrow \left(\exists x\chi \to \forall x\chi \right)\right)$$
5		df-nf	$$⊢Ⅎx\psi \leftrightarrow \left(\exists x\psi \to \forall x\psi \right)$$
6		df-nf	$$⊢Ⅎx\chi \leftrightarrow \left(\exists x\chi \to \forall x\chi \right)$$
7	4 5 6	3bitr4g	$$⊢\phi \to \left(Ⅎx\psi \leftrightarrow Ⅎx\chi \right)$$