Theorem nfcxfrd

Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016)

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

Proof

Step	Hyp	Ref	Expression
1		nfcxfr.1	$$⊢A=B$$
2		nfcxfrd.2	$$⊢\phi \to \underset{\_}{Ⅎ}xB$$
3	1	nfceqi	$$⊢\underset{\_}{Ⅎ}xA\leftrightarrow \underset{\_}{Ⅎ}xB$$
4	2 3	sylibr	$$⊢\phi \to \underset{\_}{Ⅎ}xA$$