Theorem nfcxfrd 2890

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 2888	. 2 ⊢ (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵)
4	1, 3	sylibr 234	1 ⊢ (𝜑 → Ⅎ𝑥𝐴)

Syntax hints:   → wi 4   = wceq 1540  Ⅎwnfc 2876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-nf 1784  df-cleq 2721  df-clel 2803  df-nfc 2878

This theorem is referenced by:  nfcsb1d  3881  nfcsbd  3884  nfcsbw  3885  nfifd  4514  nfunid  4873  nfopabd  5170  nfiotadw  6455  nfiotad  6457  nfriotadw  7334  nfriotad  7337  nfovd  7398  nfttrcld  9639  nfnegd  11392  nfxnegd  45410  nfintd  49635  nfiund  49636  nfiundg  49637