Theorem nfcxfrd 2891

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 2889	. 2 ⊢ (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵)
4	1, 3	sylibr 234	1 ⊢ (𝜑 → Ⅎ𝑥𝐴)

Syntax hints:   → wi 4   = wceq 1540  Ⅎwnfc 2877

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 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-nf 1784  df-cleq 2722  df-clel 2804  df-nfc 2879

This theorem is referenced by:  nfcsb1d  3887  nfcsbd  3890  nfcsbw  3891  nfifd  4521  nfunid  4880  nfopabd  5178  nfiotadw  6470  nfiotad  6472  nfriotadw  7355  nfriotad  7358  nfovd  7419  nfttrcld  9670  nfnegd  11423  nfxnegd  45444  nfintd  49666  nfiund  49667  nfiundg  49668