Theorem nfcxfrd 2217

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

Hypotheses

Ref	Expression
nfceqi.1	⊢ 𝐴 = 𝐵
nfcxfrd.2	⊢ (𝜑 → Ⅎ𝑥𝐵)

Assertion

Ref	Expression
nfcxfrd	⊢ (𝜑 → Ⅎ𝑥𝐴)

Proof of Theorem nfcxfrd

Step	Hyp	Ref	Expression
1		nfcxfrd.2	. 2 ⊢ (𝜑 → Ⅎ𝑥𝐵)
2		nfceqi.1	. . 3 ⊢ 𝐴 = 𝐵
3	2	nfceqi 2215	. 2 ⊢ (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵)
4	1, 3	sylibr 132	1 ⊢ (𝜑 → Ⅎ𝑥𝐴)

Syntax hints:   → wi 4   = wceq 1284  Ⅎwnfc 2206

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-cleq 2074  df-clel 2077  df-nfc 2208

This theorem is referenced by:  nfcsb1d  2936  nfcsbd  2939  nfifd  3376  nfunid  3608  nfiotadxy  4890  nfriotadxy  5496  nfovd  5554  nfnegd  7304