Theorem nfcxfrd 2223

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 2221	. 2 ⊢ (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵)
4	1, 3	sylibr 132	1 ⊢ (𝜑 → Ⅎ𝑥𝐴)

Syntax hints:   → wi 4   = wceq 1287  Ⅎwnfc 2212

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 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-17 1462  ax-ial 1470  ax-ext 2067

This theorem depends on definitions:  df-bi 115  df-nf 1393  df-cleq 2078  df-clel 2081  df-nfc 2214

This theorem is referenced by:  nfcsb1d  2950  nfcsbd  2953  nfifd  3404  nfunid  3643  nfiotadxy  4949  nfriotadxy  5577  nfovd  5635  nfnegd  7622