Theorem nfcxfrd 2954

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

Hypotheses

Ref	Expression
nfcxfr.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		nfcxfr.1	. . 3 ⊢ 𝐴 = 𝐵
3	2	nfceqi 2952	. 2 ⊢ (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵)
4	1, 3	sylibr 237	1 ⊢ (𝜑 → Ⅎ𝑥𝐴)

Syntax hints:   → wi 4   = wceq 1538  Ⅎwnfc 2936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-cleq 2791  df-clel 2870  df-nfc 2938

This theorem is referenced by:  nfcsb1d  3850  nfcsbd  3853  nfcsbw  3854  nfifd  4453  nfunid  4806  nfiotadw  6286  nfiotad  6288  nfriotadw  7101  nfriotad  7104  nfovd  7164  nfnegd  10870  nfxnegd  42078  nfintd  45203  nfiund  45204  nfiundg  45205