Theorem nfceqdf

Description: An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016) Avoid ax-8 and df-clel . (Revised by WL and SN, 23-Aug-2024)

	Ref	Expression
Hypotheses	nfceqdf.1	$$⊢Ⅎx\phi$$
	nfceqdf.2	$$⊢\phi \to A=B$$
Assertion	nfceqdf	$$⊢\phi \to \left(\underset{\_}{Ⅎ}xA\leftrightarrow \underset{\_}{Ⅎ}xB\right)$$

Proof

Step	Hyp	Ref	Expression
1		nfceqdf.1	$$⊢Ⅎx\phi$$
2		nfceqdf.2	$$⊢\phi \to A=B$$
3		eleq2w2	$$⊢A=B\to \left(y\in A\leftrightarrow y\in B\right)$$
4	2 3	syl	$$⊢\phi \to \left(y\in A\leftrightarrow y\in B\right)$$
5	1 4	nfbidf	$$⊢\phi \to \left(Ⅎxy\in A\leftrightarrow Ⅎxy\in B\right)$$
6	5	albidv	$$⊢\phi \to \left(\forall yℲxy\in A\leftrightarrow \forall yℲxy\in B\right)$$
7		df-nfc	$$⊢\underset{\_}{Ⅎ}xA\leftrightarrow \forall yℲxy\in A$$
8		df-nfc	$$⊢\underset{\_}{Ⅎ}xB\leftrightarrow \forall yℲxy\in B$$
9	6 7 8	3bitr4g	$$⊢\phi \to \left(\underset{\_}{Ⅎ}xA\leftrightarrow \underset{\_}{Ⅎ}xB\right)$$