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	⊢ Ⅎ 𝑥 𝜑
	nfcxfrdf.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	nfcxfrdf.2	⊢ ( 𝜑 → Ⅎ 𝑥 𝐵 )
Assertion	nfcxfrdf	⊢ ( 𝜑 → Ⅎ 𝑥 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		nfcxfrdf.0	⊢ Ⅎ 𝑥 𝜑
2		nfcxfrdf.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
3		nfcxfrdf.2	⊢ ( 𝜑 → Ⅎ 𝑥 𝐵 )
4	1 2	nfceqdf	⊢ ( 𝜑 → ( Ⅎ 𝑥 𝐴 ↔ Ⅎ 𝑥 𝐵 ) )
5	3 4	mpbird	⊢ ( 𝜑 → Ⅎ 𝑥 𝐴 )