Theorem nfcxfrdf

Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by NM, 19-Nov-2020)

	Ref	Expression
Hypotheses	nfcxfrdf.0	$$⊢Ⅎx\phi$$
	nfcxfrdf.1	$$⊢\phi \to A=B$$
	nfcxfrdf.2	$$⊢\phi \to \underset{\_}{Ⅎ}xB$$
Assertion	nfcxfrdf	$$⊢\phi \to \underset{\_}{Ⅎ}xA$$

Proof

Step	Hyp	Ref	Expression
1		nfcxfrdf.0	$$⊢Ⅎx\phi$$
2		nfcxfrdf.1	$$⊢\phi \to A=B$$
3		nfcxfrdf.2	$$⊢\phi \to \underset{\_}{Ⅎ}xB$$
4	1 2	nfceqdf	$$⊢\phi \to \left(\underset{\_}{Ⅎ}xA\leftrightarrow \underset{\_}{Ⅎ}xB\right)$$
5	3 4	mpbird	$$⊢\phi \to \underset{\_}{Ⅎ}xA$$