Theorem nfcxfrd 2218

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 2216	. 2 ⊢ (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵)
4	1, 3	sylibr 132	1 ⊢ (𝜑 → Ⅎ𝑥𝐴)

Syntax hints:   → wi 4   = wceq 1285  Ⅎwnfc 2207

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 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-cleq 2075  df-clel 2078  df-nfc 2209

This theorem is referenced by:  nfcsb1d  2937  nfcsbd  2940  nfifd  3384  nfunid  3616  nfiotadxy  4900  nfriotadxy  5507  nfovd  5565  nfnegd  7371